Adding unfinished-but-working siteswap demo, copied from an ancient perforce changelist.

samples/siteswap/math.js

+// @@REWRITE(insert js-copyright)
+// @@REWRITE(delete-start)
+// Copyright 2009 Google Inc.  All Rights Reserved
+// @@REWRITE(delete-end)
+
+/**
+ * @fileoverview This file contains all the math for the siteswap animator.  It
+ * handles all of the site-swap-related stuff [converting a sequence of integers
+ * into a more-useful representation of a pattern, pattern validation, etc.] as
+ * well as all the physics used for the simulation.
+ */
+
+/**
+ * This is a container class that holds the coefficients of an equation
+ * describing the motion of an object.
+ * The basic equation is:
+ *   f(x) := a t^2 + b t + c + d sin (f t) + e cos (f t).
+ * However, sometimes we LERP between that function and this one:
+ *   g(x) := lA t^2  + lB t + lC
+ * lerpRate [so far] is always either 1 [LERP from f to g over 1 beat] or -1,
+ * [LERP from g to f over one beat].
+ *
+ * Just plug in t to evaluate the equation.  There's no JavaScript function to
+ * do this because it's always done on the GPU.
+ *
+ * @constructor
+ */
+EquationCoefficients = function(a, b, c, d, e, f, lA, lB, lC, lerpRate) {
+  assert(!isNaN(a) && !isNaN(b) && !isNaN(c));
+  d = d || 0;
+  e = e || 0;
+  f = f || 0;
+  assert(!isNaN(d) && !isNaN(e) && !isNaN(f));
+  lA = lA || 0;
+  lB = lB || 0;
+  lC = lC || 0;
+  assert(!isNaN(lA) && !isNaN(lB) && !isNaN(lC));
+  lerpRate = lerpRate || 0;
+  this.a = a;
+  this.b = b;
+  this.c = c;
+  this.d = d;
+  this.e = e;
+  this.f = f;
+  this.lA = lA;
+  this.lB = lB;
+  this.lC = lC;
+  this.lerpRate = lerpRate;
+}
+
+/**
+ * Create a new equation that's equivalent to this equation's coefficients a-f
+ * with a LERP to the polynomial portion of the supplied equation.
+ * @param {!EquationCoefficients} eqn the source of coefficients.
+ * @param {!number} lerpRate the rate and direction of the LERP; positive for
+ *     "from this equation to the new one" and vice-versa.
+ * @return {!EquationCoefficients} a new set of coefficients.
+ */
+EquationCoefficients.prototype.lerpIn = function(eqn, lerpRate) {
+  assert(!this.lerpRate);
+  return new EquationCoefficients(this.a, this.b, this.c, this.d, this.e,
+      this.f, eqn.a, eqn.b, eqn.c, lerpRate);
+};
+
+/**
+ * Convert the EquationCoefficients to a string for debugging.
+ * @return {String} debugging output.
+ */
+EquationCoefficients.prototype.toString = function() {
+  return 'F(t) := ' + this.a.toFixed(2) + ' t^2 + ' + this.b.toFixed(2) +
+      ' t + ' + this.c.toFixed(2) + ' + ' +
+      this.d.toFixed(2) + ' sin(' + this.f.toFixed(2) + ' t) + ' +
+      this.e.toFixed(2) + ' cos(' + this.f.toFixed(2) + ' t) + LERP(' +
+      this.lerpRate.toFixed(2) + ') of ' +
+      this.lA.toFixed(2) + ' t^2 + ' + this.lB.toFixed(2) +
+      ' t + ' + this.lC.toFixed(2);
+};
+
+/**
+ * A set of equations which describe the motion of an object over time.
+ * The three equations each supply one dimension of the motion, and the curve is
+ * valid from startTime to startTime + duration.
+ * @param {!number} startTime the initial time at which the curve is valid.
+ * @param {!number} duration how long [in beats] the curve is valid.
+ * @param {!EquationCoefficients} xEqn the equation for motion in x.
+ * @param {!EquationCoefficients} yEqn the equation for motion in y.
+ * @param {!EquationCoefficients} zEqn the equation for motion in z.
+ * @constructor
+ */
+Curve = function(startTime, duration, xEqn, yEqn, zEqn) {
+  this.startTime = startTime;
+  this.duration = duration;
+  this.xEqn = xEqn;
+  this.yEqn = yEqn;
+  this.zEqn = zEqn;
+}
+
+/**
+ * Convert the Curve to a string for debugging.
+ * @return {String} debugging output.
+ */
+Curve.prototype.toString = function() {
+  var s = 'startTime: ' + this.startTime + '\n';
+  s += 'duration: ' + this.duration + '\n';
+  s += this.xEqn + '\n';
+  s += this.yEqn + '\n';
+  s += this.zEqn + '\n';
+  return s;
+};
+
+/**
+ * Modify this curve's coefficients to include a LERP to the polynomial
+ * portion of the supplied curve.
+ * @param {!Curve} curve the source of coefficients.
+ * @param {!number} lerpRate the rate and direction of the LERP; positive for
+ *     "from this equation to the new one" and vice-versa.
+ * @return {!Curve} a new curve.
+ */
+Curve.prototype.lerpIn = function(curve, lerpRate) {
+  assert(this.startTime == curve.startTime);
+  assert(this.duration == curve.duration);
+  var xEqn = this.xEqn.lerpIn(curve.xEqn, lerpRate);
+  var yEqn = this.yEqn.lerpIn(curve.yEqn, lerpRate);
+  var zEqn = this.zEqn.lerpIn(curve.zEqn, lerpRate);
+  return new Curve(this.startTime, this.duration, xEqn, yEqn, zEqn);
+};
+
+/**
+ * Produce a set of polynomial coefficients that describe linear motion between
+ * two points in 1 dimension.
+ * @param {!number} startPos the starting position.
+ * @param {!number} endPos the ending position.
+ * @param {!number} duration how long the motion takes.
+ * @return {!EquationCoefficients} the equation for the motion.
+ */
+Curve.computeLinearCoefficients = function(startPos, endPos, duration) {
+  return new EquationCoefficients(
+      0, (endPos - startPos) / duration, startPos);
+}
+
+var GRAVITY = 1; // Higher means higher throws for the same duration.
+/**
+ * Produce a set of polynomial coefficients that describe parabolic motion
+ * between two points in 1 dimension.
+ * @param {!number} startPos the starting position.
+ * @param {!number} endPos the ending position.
+ * @param {!number} duration how long the motion takes.
+ * @return {!EquationCoefficients} the equation for the motion.
+ */
+Curve.computeParabolicCoefficients = function(startPos, endPos, duration) {
+  var dY = endPos - startPos;
+  return new EquationCoefficients(-GRAVITY / 2,
+                                  dY / duration + GRAVITY * duration / 2,
+                                  startPos);
+}
+
+/**
+ * Compute the curve taken by a ball given its throw and catch positions, the
+ * time it was thrown, and how long it stayed in the air.
+ *
+ * We use duration rather than throwTime and catchTime because, what
+ * with the modular arithmetic used in our records, catchTime might be before
+ * throwTime, and in some representations the pattern could wrap around a few
+ * times while the ball's in the air.  When the parabola computed here is used,
+ * time must be supplied as an offset from the time of the throw, and must of
+ * course not wrap at all.  That is, these coefficients work for f(0) ==
+ * throwPos, f(duration) == catchPos.
+ *
+ * We treat the y axis as vertical and thus affected by gravity.
+ *
+ * @param {!EquationCoefficients} throwPos
+ * @param {!EquationCoefficients} catchPos
+ * @param {!number} startTime
+ * @param {!number} duration
+ * @return {!Curve}
+ */
+Curve.computeThrowCurve = function(throwPos, catchPos, startTime, duration) {
+  var xEqn = Curve.computeLinearCoefficients(throwPos.x, catchPos.x, duration);
+  var yEqn = Curve.computeParabolicCoefficients(throwPos.y, catchPos.y,
+      duration);
+  var zEqn = Curve.computeLinearCoefficients(throwPos.z, catchPos.z, duration);
+  return new Curve(startTime, duration, xEqn, yEqn, zEqn);
+}
+
+/**
+ * Compute a straight line Curve given start and end positions, the start time,
+ * and the duration of the motion.
+ *
+ * @param {!EquationCoefficients} startPos
+ * @param {!EquationCoefficients} endPos
+ * @param {!number} startTime
+ * @param {!number} duration
+ * @return {!Curve}
+ */
+Curve.computeStraightLineCurve =
+    function(startPos, endPos, startTime, duration) {
+  var xEqn = Curve.computeLinearCoefficients(startPos.x, endPos.x, duration);
+  var yEqn = Curve.computeLinearCoefficients(startPos.y, endPos.y, duration);
+  var zEqn = Curve.computeLinearCoefficients(startPos.z, endPos.z, duration);
+  return new Curve(startTime, duration, xEqn, yEqn, zEqn);
+}
+
+/**
+ * Threshold horizontal distance below which computeCircularCurve won't bother
+ * trying to approximate a circular curve.  See the comment above
+ * computeCircularCurve for more info.
+ * @type {number}
+ */
+Curve.EPSILON = .0001;
+
+/**
+ * Compute a circular curve, used as an approximation for the motion of a hand
+ * between a catch and its following throw.
+ *
+ * Assumes a lot of stuff about this looking like a "normal" throw: the catch is
+ * moving roughly the opposite direction as the throw, the throw and catch
+ * aren't at the same place, and such.  Otherwise this looks very odd at best.
+ * This is used for the height of the curve.
+ * This produces coefficients for d sin(f t) + e cos(f t) for each of x, y, z.
+ * It produces a vertical-ish circular curve from the start to the end, going
+ * down, then up.  So if dV [the distance from the start to finish in the x-z
+ * plane, ignoring y] is less than Curve.EPSILON, it doesn't know which way down
+ * is, and it bails by returning a straight line instead.
+ */
+Curve.computeCircularCurve = function(startPos, endPos, startTime, duration) {
+  var dX = endPos.x - startPos.x;
+  var dY = endPos.y - startPos.y;
+  var dZ = endPos.z - startPos.z;
+  var dV = Math.sqrt(dX * dX + dZ * dZ);
+  if (dV < Curve.EPSILON) {
+    return Curve.computeStraightLineCurve(startPos, endPos, startTime,
+        duration);
+  }
+  var negHalfdV = -0.5 * dV;
+  var negHalfdY = -0.5 * dY;
+  var f = Math.PI / duration;
+  var yEqn = new EquationCoefficients(
+      0, 0, startPos.y + dY / 2,
+      negHalfdV, negHalfdY, f);
+  var ratio = dX / dV;
+  var xEqn = new EquationCoefficients(
+      0, 0, startPos.x + dX / 2,
+      negHalfdY * ratio, negHalfdV * ratio, f);
+  ratio = dZ / dV;
+  var zEqn = new EquationCoefficients(
+      0, 0, startPos.z + dZ / 2,
+      negHalfdY * ratio, negHalfdV * ratio, f);
+  return new Curve(startTime, duration, xEqn, yEqn, zEqn);
+}
+
+/**
+ * This is the abstract base class for an object that describes a throw, catch,
+ * or empty hand [placeholder] in a site-swap pattern.
+ * @constructor
+ */
+Descriptor = function() {
+}
+
+/**
+ * Create an otherwise-identical copy of this descriptor at a given time offset.
+ * Note that offset may put time past patternLength; the caller will have to fix
+ * this up manually.
+ * @param {number} offset how many beats to offset the new descriptor.
+ * Derived classes must override this function.
+ */
+Descriptor.prototype.clone = function(offset) {
+  throw new Error('Unimplemented.');
+};
+
+/**
+ * Generate the Curve implied by this descriptor and the supplied hand
+ * positions.
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * Derived classes must override this function.
+ */
+Descriptor.prototype.generateCurve = function(handPositions) {
+  throw new Error('Unimplemented.');
+};
+
+/**
+ * Adjust the start time of this Descriptor to be in [0, pathLength).
+ * @param {!number} pathLength the duration of a path, in beats.
+ * @return {!Descriptor} this.
+ */
+Descriptor.prototype.fixUpModPathLength = function(pathLength) {
+  this.time = this.time % pathLength;
+  return this;
+};
+
+/**
+ * This describes a throw in a site-swap pattern.
+ * @param {!number} throwNum the site-swap number of the throw.
+ * @param {!number} throwTime the time this throw occurs.
+ * @param {!number} sourceHand the index of the throwing hand.
+ * @param {!number} destHand the index of the catching hand.
+ * @constructor
+ */
+ThrowDescriptor = function(throwNum, throwTime, sourceHand, destHand) {
+  this.throwNum = throwNum;
+  this.sourceHand = sourceHand;
+  this.destHand = destHand;
+  this.time = throwTime;
+}
+
+/**
+ * This is a subclass of Descriptor.
+ */
+ThrowDescriptor.prototype = new Descriptor();
+
+/**
+ * Set up the constructor, just to be neat.
+ */
+ThrowDescriptor.prototype.constructor = ThrowDescriptor;
+
+/**
+ * We label each Descriptor subclass with a type for debugging.
+ */
+ThrowDescriptor.prototype.type = 'THROW';
+
+/**
+ * Create an otherwise-identical copy of this descriptor at a given time offset.
+ * Note that offset may put time past patternLength; the caller will have to fix
+ * this up manually.
+ * @param {number} offset how many beats to offset the new descriptor.
+ * @return {!Descriptor} the new copy.
+ */
+ThrowDescriptor.prototype.clone = function(offset) {
+  offset = offset || 0;  // Turn null into 0.
+  return new ThrowDescriptor(this.throwNum, this.time + offset,
+      this.sourceHand, this.destHand);
+};
+
+/**
+ * Convert the ThrowDescriptor to a string for debugging.
+ * @return {String} debugging output.
+ */
+ThrowDescriptor.prototype.toString = function() {
+  return '(' + this.throwNum + ' from hand ' + this.sourceHand + ' to hand ' +
+    this.destHand + ')';
+};
+
+/**
+ * Generate the Curve implied by this descriptor and the supplied hand
+ * positions.
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * @return {!Curve} the curve.
+ */
+ThrowDescriptor.prototype.generateCurve = function(handPositions) {
+  var startPos = handPositions[this.sourceHand].throwPositions[this.destHand];
+  var endPos = handPositions[this.destHand].catchPosition;
+  return Curve.computeThrowCurve(startPos, endPos, this.time,
+      this.throwNum - 1); };
+
+/**
+ * This describes a catch in a site-swap pattern.
+ * @param {!number} hand the index of the catching hand.
+ * @param {!number} sourceThrowNum the site-swap number of the preceeding throw.
+ * @param {!number} destThrowNum the site-swap number of the following throw.
+ * @param {!number} sourceHand the index of the hand throwing the source throw.
+ * @param {!number} destHand the index of the hand catching the following throw.
+ * @param {!number} catchTime the time at which the catch occurs.
+ * @constructor
+ */
+CarryDescriptor = function(hand, sourceThrowNum, destThrowNum, sourceHand,
+    destHand, catchTime) {
+  this.hand = hand;
+  this.sourceThrowNum = sourceThrowNum;
+  this.destThrowNum = destThrowNum;
+  this.sourceHand = sourceHand;
+  this.destHand = destHand;
+  this.time = catchTime;
+}
+
+/**
+ * This is a subclass of Descriptor.
+ */
+CarryDescriptor.prototype = new Descriptor();
+
+/**
+ * Set up the constructor, just to be neat.
+ */
+CarryDescriptor.prototype.constructor = CarryDescriptor;
+
+/**
+ * We label each Descriptor subclass with a type for debugging.
+ */
+CarryDescriptor.prototype.type = 'CARRY';
+
+/**
+ * Since this gets pathLength, not patternLength, we'll have to collapse sets
+ * of CarryDescriptors later, as they may be spread sparsely through the full
+ * animation and we'll only want them to be distributed over the full pattern
+ * length.  We may have dupes to throw away as well.
+ * @param {!ThrowDescriptor} inThrowDescriptor
+ * @param {!ThrowDescriptor} outThrowDescriptor
+ * @param {!number} pathLength
+ * @return {!CarryDescriptor}
+ */
+CarryDescriptor.fromThrowDescriptors = function(inThrowDescriptor,
+    outThrowDescriptor, pathLength) {
+  assert(inThrowDescriptor.destHand == outThrowDescriptor.sourceHand);
+  assert((inThrowDescriptor.time + inThrowDescriptor.throwNum) %
+      pathLength == outThrowDescriptor.time);
+  return new CarryDescriptor(inThrowDescriptor.destHand,
+      inThrowDescriptor.throwNum, outThrowDescriptor.throwNum,
+      inThrowDescriptor.sourceHand, outThrowDescriptor.destHand,
+      (outThrowDescriptor.time + pathLength - 1) % pathLength);
+};
+
+/**
+ * Create an otherwise-identical copy of this descriptor at a given time offset.
+ * Note that offset may put time past patternLength; the caller will have to fix
+ * this up manually.
+ * @param {number} offset how many beats to offset the new descriptor.
+ * @return {!Descriptor} the new copy.
+ */
+CarryDescriptor.prototype.clone = function(offset) {
+  offset = offset || 0;  // Turn null into 0.
+  return new CarryDescriptor(this.hand, this.sourceThrowNum,
+      this.destThrowNum, this.sourceHand, this.destHand, this.time + offset);
+};
+
+/**
+ * Convert the CarryDescriptor to a string for debugging.
+ * @return {String} debugging output.
+ */
+CarryDescriptor.prototype.toString = function() {
+  return 'time: ' + this.time + ' (hand ' + this.hand + ' catches ' +
+    this.sourceThrowNum + ' from hand ' + this.sourceHand + ' then throws ' +
+    this.destThrowNum + ' to hand ' + this.destHand + ')';
+};
+
+/**
+ * Test if this CarryDescriptor is equivalent to another, mod patternLength.
+ * @param {!CarryDescriptor} cd the other CarryDescriptor.
+ * @param {!number} patternLength the length of the pattern.
+ * @return {!bool}
+ */
+CarryDescriptor.prototype.equalsWithMod = function(cd, patternLength) {
+  if (!(cd instanceof CarryDescriptor)) {
+    return false;
+  }
+  if (this.hand != cd.hand) {
+    return false;
+  }
+  if (this.sourceThrowNum != cd.sourceThrowNum) {
+    return false;
+  }
+  if (this.destThrowNum != cd.destThrowNum) {
+    return false;
+  }
+  if (this.sourceHand != cd.sourceHand) {
+    return false;
+  }
+  if (this.destHand != cd.destHand) {
+    return false;
+  }
+  if (this.time % patternLength != cd.time % patternLength) {
+    return false;
+  }
+  return true;
+};
+
+/**
+ * Generate the Curve implied by this descriptor and the supplied hand
+ * positions.
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * @return {!Curve} the curve.
+ */
+CarryDescriptor.prototype.generateCurve = function(handPositions) {
+  var startPos = handPositions[this.hand].catchPosition;
+  var endPos = handPositions[this.hand].throwPositions[this.destHand];
+  return Curve.computeCircularCurve(startPos, endPos, this.time, 1);
+};
+
+/**
+ * This describes a carry of a "1" in a site-swap pattern.
+ * The flags isThrow and isCatch tell whether this is the actual 1 [isThrow] or
+ * the carry that receives the handoff [isCatch].  It's legal for both to be
+ * true, which happens when there are two 1s in a row.
+ * @param {!number} sourceThrowNum the site-swap number of the prev throw
+ * [including this one if isCatch].
+ * @param {!number} sourceHand the index of the hand throwing sourceThrowNum.
+ * @param {!number} destThrowNum the site-swap number of the next throw
+ * [including this one if isThrow].
+ * @param {!number} destHand the index of the hand catching destThrowNum.
+ * @param {!number} hand the index of the hand doing this carry.
+ * @param {!number} time the time at which the carry starts.
+ * @param {!bool} isThrow whether this is a 1.
+ * @param {!bool} isCatch whether this is the carry after a 1.
+ * @constructor
+ */
+CarryOneDescriptor = function(sourceThrowNum, sourceHand, destThrowNum,
+    destHand, hand, time, isThrow, isCatch) {
+  // It's possible to have !isCatch with sourceThrowNum == 1 temporarily, if we
+  // just haven't handled that 1 yet [we're doing the throw of this one, and
+  // will later get to the previous one, due to wraparound], and vice-versa.
+  assert(isThrow || (sourceThrowNum == 1));
+  assert(isCatch || (destThrowNum == 1));
+  this.sourceThrowNum = sourceThrowNum;
+  this.sourceHand = sourceHand;
+  this.destHand = destHand;
+  this.destThrowNum = destThrowNum;
+  this.hand = hand;
+  this.time = time;
+  this.isThrow = isThrow;
+  this.isCatch = isCatch;
+  return this;
+}
+
+/**
+ * This is a subclass of Descriptor.
+ */
+CarryOneDescriptor.prototype = new Descriptor();
+
+/**
+ * Set up the constructor, just to be neat.
+ */
+CarryOneDescriptor.prototype.constructor = CarryOneDescriptor;
+
+/**
+ * We label each Descriptor subclass with a type for debugging.
+ */
+CarryOneDescriptor.prototype.type = 'CARRY_ONE';
+
+/**
+ * Create a pair of CarryOneDescriptors to describe the carry that is a throw of
+ * 1.  A 1 spends all its time being carried, so these two carries surrounding
+ * it represent [and therefore don't have] a throw between them.
+ * Prev and post are generally the ordinary CarryDescriptors surrounding the
+ * throw of 1 that we're trying to implement.  However, they could each [or
+ * both] independently be CarryOneDescriptors implementing other 1 throws.
+ * @param {!Descriptor} prev the carry descriptor previous to the 1.
+ * @param {!Descriptor} post the carry descriptor subsequent to the 1.
+ * @return {!Array.CarryOneDescriptor} a pair of CarryOneDescriptors.
+ */
+CarryOneDescriptor.getDescriptorPair = function(prev, post) {
+  assert(prev instanceof CarryDescriptor || prev instanceof CarryOneDescriptor);
+  assert(post instanceof CarryDescriptor || post instanceof CarryOneDescriptor);
+  assert(prev.destHand == post.hand);
+  assert(prev.hand == post.sourceHand);
+  var newPrev;
+  var newPost;
+  if (prev instanceof CarryOneDescriptor) {
+    assert(prev.isCatch && !prev.isThrow);
+    newPrev = prev;
+    newPrev.isThrow = true;
+    assert(newPrev.destHand == post.hand);
+  } else {
+    newPrev = new CarryOneDescriptor(prev.sourceThrowNum, prev.sourceHand, 1,
+        post.hand, prev.hand, prev.time, true, false);
+  }
+  if (post instanceof CarryOneDescriptor) {
+    assert(post.isThrow && !post.isCatch);
+    newPost = post;
+    newPost.isCatch = true;
+    assert(newPost.sourceHand == prev.hand);
+    assert(newPost.sourceThrowNum == 1);
+  } else {
+    newPost = new CarryOneDescriptor(1, prev.hand, post.destThrowNum,
+        post.destHand, post.hand, post.time, false, true);
+  }
+  return [newPrev, newPost];
+};
+
+/**
+ * Convert the CarryOneDescriptor to a string for debugging.
+ * @return {String} debugging output.
+ */
+CarryOneDescriptor.prototype.toString = function() {
+  var s;
+  if (this.isThrow) {
+    s = 'Hand ' + this.hand + ' catches a ' + this.sourceThrowNum + ' from ' +
+        this.sourceHand + ' at time ' + this.time + ' and then passes a 1 to ' +
+        this.destHand + '.';
+  } else {
+    assert(this.isCatch && this.sourceThrowNum == 1);
+    s = 'Hand ' + this.hand + ' catches a 1 from ' + this.sourceHand +
+        ' at time ' + this.time + ' and then passes a ' + this.destThrowNum +
+        ' to ' + this.destHand + '.';
+  }
+  return s;
+};
+
+/**
+ * Compute the curve taken by a ball during the carry representing a 1, as long
+ * as it's not both a catch and a throw of a 1, which is handled elsewhere.
+ * It's either a LERP from a circular curve [a catch of a throw > 1] to a
+ * straight line to the handoff point [for isThrow] or a LERP from a straight
+ * line from the handoff to a circular curve for the next throw > 1 [for
+ * isCatch].
+ *
+ * @param {!EquationCoefficients} catchPos
+ * @param {!EquationCoefficients} throwPos
+ * @param {!EquationCoefficients} handoffPos
+ * @param {!number} startTime
+ * @param {!bool} isCatch whether this is the carry after a 1.
+ * @param {!bool} isThrow whether this is a 1.
+ * @return {!Curve}
+ */
+Curve.computeCarryOneCurve = function(catchPos, throwPos, handoffPos, startTime,
+    isCatch, isThrow) {
+  assert(!isCatch != !isThrow);
+  var curve = Curve.computeCircularCurve(catchPos, throwPos, startTime, 1);
+  var curve2 = Curve.computeStraightLineCurve(handoffPos, handoffPos,
+      startTime, 1);
+  return curve.lerpIn(curve2, isThrow ? 1 : -1);
+}
+
+/**
+ * Compute the curve taken by a ball during the carry representing a 1 that is
+ * both the catch of one 1 and the immediately-following throw of another 1.
+ *
+ * @param {!EquationCoefficients} leadingHandoffPos
+ * @param {!EquationCoefficients} trailingHandoffPos
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * @param {!number} hand
+ * @param {!number} time the time at which the first 1's catch takes place.
+ * @return {!Curve}
+ */
+Curve.computeConsecutiveCarryOneCurve = function(leadingHandoffPos,
+    trailingHandoffPos, handPositions, hand, time) {
+  var curve = Curve.computeStraightLineCurve(leadingHandoffPos,
+      handPositions[hand].basePosition, time, 1);
+  var curve2 =
+    Curve.computeStraightLineCurve(handPositions[hand].basePosition,
+        trailingHandoffPos, time, 1);
+  return curve.lerpIn(curve2, 1);
+}
+
+/**
+ * Generate the Curve implied by this descriptor and the supplied hand
+ * positions.
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * @return {!Curve} the curve.
+ */
+CarryOneDescriptor.prototype.generateCurve = function(handPositions) {
+  var leadingHandoffPos, trailingHandoffPos;
+  if (this.isCatch) {
+    var p0 = handPositions[this.hand].basePosition;
+    var p1 = handPositions[this.sourceHand].basePosition;
+    handoffPos = leadingHandoffPos = p0.add(p1).scale(0.5);
+  }
+  if (this.isThrow) {
+    var p0 = handPositions[this.hand].basePosition;
+    var p1 = handPositions[this.destHand].basePosition;
+    handoffPos = trailingHandoffPos = p0.add(p1).scale(0.5);
+  }
+  if (!this.isCatch || !this.isThrow) {
+    return Curve.computeCarryOneCurve(handPositions[this.hand].catchPosition,
+        handPositions[this.hand].throwPositions[this.destHand], handoffPos,
+        this.time, this.isCatch, this.isThrow);
+  } else {
+    return Curve.computeConsecutiveCarryOneCurve(leadingHandoffPos,
+        trailingHandoffPos, handPositions, this.hand, this.time);
+  }
+};
+
+/**
+ * Create an otherwise-identical copy of this descriptor at a given time offset.
+ * Note that offset may put time past patternLength; the caller will have to fix
+ * this up manually.
+ * @param {number} offset how many beats to offset the new descriptor.
+ * @return {!Descriptor} the new copy.
+ */
+CarryOneDescriptor.prototype.clone = function(offset) {
+  offset = offset || 0;  // Turn null into 0.
+  return new CarryOneDescriptor(this.sourceThrowNum, this.sourceHand,
+      this.destThrowNum, this.destHand, this.hand, this.time + offset,
+      this.isThrow, this.isCatch);
+};
+
+/**
+ * Test if this CarryOneDescriptor is equivalent to another, mod patternLength.
+ * @param {!CarryOneDescriptor} cd the other CarryOneDescriptor.
+ * @param {!number} patternLength the length of the pattern.
+ * @return {!bool}
+ */
+CarryOneDescriptor.prototype.equalsWithMod = function(cd, patternLength) {
+  if (!(cd instanceof CarryOneDescriptor)) {
+    return false;
+  }
+  if (this.hand != cd.hand) {
+    return false;
+  }
+  if (this.sourceThrowNum != cd.sourceThrowNum) {
+    return false;
+  }
+  if (this.destThrowNum != cd.destThrowNum) {
+    return false;
+  }
+  if (this.sourceHand != cd.sourceHand) {
+    return false;
+  }
+  if (this.destHand != cd.destHand) {
+    return false;
+  }
+  if (this.isCatch != cd.isCatch) {
+    return false;
+  }
+  if (this.isThrow != cd.isThrow) {
+    return false;
+  }
+  if (this.time % patternLength != cd.time % patternLength) {
+    return false;
+  }
+  return true;
+};
+
+/**
+ * This describes an empty hand in a site-swap pattern.
+ * @param {!Descriptor} cd0 the CarryDescriptor or CarryOneDescriptor describing
+ * this hand immediately before it was emptied.
+ * @param {!Descriptor} cd1 the CarryDescriptor or CarryOneDescriptor describing
+ * this hand immediately after it's done being empty.
+ * @param {!number} patternLength the length of the pattern.
+ * @constructor
+ */
+EmptyHandDescriptor = function(cd0, cd1, patternLength) {
+  assert(cd0.hand == cd1.hand);
+  this.hand = cd0.hand;
+  this.prevThrowDest = cd0.destHand;
+  this.sourceThrowNum = cd0.destThrowNum;
+  this.nextCatchSource = cd1.sourceHand;
+  this.destThrowNum = cd1.sourceThrowNum;
+  // This code assumes that each CarryDescriptor and CarryOneDescriptor always
+  // has a duration of 1 beat.  If we want to be able to allow long-held balls
+  // [instead of thrown twos, for example], we'll have to fix that here and a
+  // number of other places.
+  this.time = (cd0.time + 1) % patternLength;
+  this.duration = cd1.time - this.time;
+  if (this.duration < 0) {
+    this.duration += patternLength;
+    assert(this.duration > 0);
+  }
+}
+
+/**
+ * This is a subclass of Descriptor.
+ */
+EmptyHandDescriptor.prototype = new Descriptor();
+
+/**
+ * Set up the constructor, just to be neat.
+ */
+EmptyHandDescriptor.prototype.constructor = EmptyHandDescriptor;
+
+/**
+ * We label each Descriptor subclass with a type for debugging.
+ */
+EmptyHandDescriptor.prototype.type = 'EMPTY';
+
+/**
+ * Convert the EmptyHandDescriptor to a string for debugging.
+ * @return {String} debugging output.
+ */
+EmptyHandDescriptor.prototype.toString = function() {
+  return 'time: ' + this.time + ' for ' + this.duration + ' (hand ' +
+      this.hand + ', after throwing a ' + this.sourceThrowNum + ' to hand ' +
+      this.prevThrowDest + ' then catches a ' + this.destThrowNum +
+      ' from hand ' + this.nextCatchSource + ')';
+};
+
+/**
+ * Generate the Curve implied by this descriptor and the supplied hand
+ * positions.
+ * @param {!Array.HandPositionRecord} handPositions where the hands will be.
+ * @return {!Curve} the curve.
+ */
+EmptyHandDescriptor.prototype.generateCurve = function(handPositions) {
+  var startPos, endPos;
+  if (this.sourceThrowNum == 1) {
+    var p0 = handPositions[this.hand].basePosition;
+    var p1 = handPositions[this.prevThrowDest].basePosition;
+    startPos = p0.add(p1).scale(0.5);
+  } else {
+    startPos = handPositions[this.hand].throwPositions[this.prevThrowDest];
+  }
+  if (this.destThrowNum == 1) {
+    var p0 = handPositions[this.hand].basePosition;
+    var p1 = handPositions[this.nextCatchSource].basePosition;
+    endPos = p0.add(p1).scale(0.5);
+  } else {
+    endPos = handPositions[this.hand].catchPosition;
+  }
+  // TODO: Replace with a good empty-hand curve.
+  return Curve.computeStraightLineCurve(startPos, endPos, this.time,
+      this.duration);
+};
+
+/**
+ * A series of descriptors that describes the full path of an object during a
+ * pattern.
+ * @param {!Array.Descriptor} descriptors all descriptors for the object.
+ * @param {!number} pathLength the length of the path in beats.
+ * @constructor
+ */
+Path = function(descriptors, pathLength) {
+  this.descriptors = descriptors;
+  this.pathLength = pathLength;
+}
+
+/**
+ * Create a Path representing a ball, filling in the gaps between the throws
+ * with carry descriptors.  Since it's a ball's path, there are no
+ * EmptyHandDescriptors in the output.
+ * @param {!Array.ThrowDescriptor} throwDescriptors the ball's part of the
+ * pattern.
+ * @param {!number} pathLength the length of the pattern in beats.
+ * @return {!Path} the ball's full path.
+ */
+Path.ballPathFromThrowDescriptors = function(throwDescriptors, pathLength) {
+  return new Path(
+      Path.createDescriptorList(throwDescriptors, pathLength), pathLength);
+};
+
+/**
+ * Create the sequence of ThrowDescriptors, CarryDescriptors, and
+ * CarryOneDescriptor describing the path of a ball through a pattern.
+ * A sequence such as (h j k) generally maps to an alternating series of throw
+ * and carry descriptors [Th Chj Tj Cjk Tk Ck? ...].  However, when j is a 1,
+ * you remove the throw descriptor and modify the previous and subsequent carry
+ * descriptors, since the throw descriptor has zero duration and the carry
+ * descriptors need to take into account the handoff.
+ * @param {!Array.ThrowDescriptor} throwDescriptors the ball's part of the
+ * pattern.
+ * @param {!number} pathLength the length of the pattern in beats.
+ * @return {!Array.Descriptor} the full set of descriptors for the ball.
+ */
+Path.createDescriptorList = function(throwDescriptors, pathLength) {
+  var descriptors = [];
+  var prevThrow;
+  for (var index in throwDescriptors) {
+    var td = throwDescriptors[index];
+    if (prevThrow) {
+      descriptors.push(
+          CarryDescriptor.fromThrowDescriptors(prevThrow, td, pathLength));
+    } // Else it's handled after the loop.
+    descriptors.push(td);
+    prevThrow = td;
+  }
+  descriptors.push(
+      CarryDescriptor.fromThrowDescriptors(prevThrow, throwDescriptors[0],
+        pathLength));
+  // Now post-process to take care of throws of 1.  It's easier to do it here
+  // than during construction since we can now assume that the previous and
+  // subsequent carry descriptors are already in place [modulo pathLength].
+  for (var i = 0; i < descriptors.length; ++i) {
+    var descriptor = descriptors[i];
+    if (descriptor instanceof ThrowDescriptor) {
+      if (descriptor.throwNum == 1) {
+        var prevIndex = (i + descriptors.length - 1) % descriptors.length;
+        var postIndex = (i + 1) % descriptors.length;
+        var replacements = CarryOneDescriptor.getDescriptorPair(
+            descriptors[prevIndex], descriptors[postIndex]);
+        descriptors[prevIndex] = replacements[0];
+        descriptors[postIndex] = replacements[1];
+        descriptors.splice(i, 1);
+        // We've removed a descriptor from the array, but since we can never
+        // have 2 ThrowDescriptors in a row, we don't need to decrement i.
+      }
+    }
+  }
+  return descriptors;
+};
+
+/**
+ * Convert the Path to a string for debugging.
+ * @return {String} debugging output.
+ */
+Path.prototype.toString = function() {
+  var ret = 'pathLength is ' + this.pathLength + '; [';
+  for (var index in this.descriptors) {
+    ret += this.descriptors[index].toString();
+  }
+  ret += ']';
+  return ret;
+};
+
+/**
+ * Create an otherwise-identical copy of this path at a given time offset.
+ * Note that offset may put time references in the Path past the length of the
+ * pattern.  The caller must fix this up manually.
+ * @param {number} offset how many beats to offset the new Path.
+ * @return {!Path} the new copy.
+ */
+Path.prototype.clone = function(offset) {
+  offset = offset || 0;  // Turn null into 0.
+  var descriptors = [];
+  for (var index in this.descriptors) {
+    descriptors.push(this.descriptors[index].clone(offset));
+  }
+  return new Path(descriptors, this.pathLength);
+};
+
+/**
+ * Adjust the start time of all descriptors to be in [0, pathLength) via modular
+ * arithmetic.  Reorder the array such that they're sorted in increasing order
+ * of time.
+ * @return {!Path} this.
+ */
+Path.prototype.fixUpModPathLength = function() {
+  var splitIndex;
+  var prevTime = 0;
+  for (var index in this.descriptors) {
+    var d = this.descriptors[index];
+    d.fixUpModPathLength(this.pathLength);
+    if (d.time < prevTime) {
+      assert(null == splitIndex);
+      splitIndex = index; // From here to the end should move to the start.
+    }
+    prevTime = d.time;
+  }
+  if (null != splitIndex) {
+    var temp = this.descriptors.slice(splitIndex);
+    this.descriptors.length = splitIndex;
+    this.descriptors = temp.concat(this.descriptors);
+  }
+  return this;
+};
+
+/**
+ * Take a standard asynch siteswap pattern [expressed as an array of ints] and
+ * a number of hands, and expand it into a 2D grid of ThrowDescriptors with one
+ * row per hand.
+ * Non-asynch patterns are more complicated, since their linear forms aren't
+ * fully-specified, so we don't handle them here.
+ * You'll want to expand your pattern to the LCM of numHands and minimal pattern
+ * length before calling this.
+ * The basic approach doesn't really work for one-handed patterns.  It ends up
+ * with catches and throws happening at the same time [having removed all
+ * empty-hand time in between them].  To fix this, we double all throw heights
+ * and space them out, as if doing a two-handed pattern with all zeroes from the
+ * other hand.  Yes, this points out that the overall approach we're taking is a
+ * bit odd [since you end up with hands empty for time proportional to the
+ * number of hands], but you have to make some sort of assumptions to generalize
+ * siteswaps to N hands, and that's what I chose.
+ * @param {!Array.number} pattern an asynch siteswap pattern.
+ * @param {!number} numHands the number of hands.
+ * @return {!Array.Array.ThrowDescriptor} the expanded pattern.
+ */
+function expandPattern(pattern, numHands) {
+  var fullPattern = [];
+  assert(numHands > 0);
+  if (numHands == 1) {
+    numHands = 2;
+    var temp = [];
+    for (var i = 0; i < pattern.length; ++i) {
+      temp[2 * i] = 2 * pattern[i];
+      temp[2 * i + 1] = 0;
+    }
+    pattern = temp;
+  }
+  for (var hand = 0; hand < numHands; ++hand) {
+    fullPattern[hand] = [];
+  }
+  for (var time = 0; time < pattern.length; ++time) {
+    for (var hand = 0; hand < numHands; ++hand) {
+      var t;
+      if (hand == time % numHands) {
+        t = new ThrowDescriptor(pattern[time], time, hand,
+            (hand + pattern[time]) % numHands);
+      } else {
+        // These are ignored during analysis, so they don't appear in BallPaths.
+        t = new ThrowDescriptor(0, time, hand, hand);
+      }
+      fullPattern[hand].push(t);
+    }
+  }
+  return fullPattern;
+}
+
+// TODO: Wrap the final pattern in a class, then make the remaining few global
+// functions be members of that class to clean up the global namespace.
+
+/**
+ * Given a valid site-swap for a nonzero number of balls, stored as an expanded
+ * pattern array-of-arrays, with pattern length the LCM of hands and minimal
+ * pattern length, produce Paths for all the balls.
+ * @param {!Array.Array.ThrowDescriptor} pattern a valid pattern.
+ * @return {!Array.Path} the paths of all the balls.
+ */
+function generateBallPaths(pattern) {
+  var numHands = pattern.length;
+  assert(numHands > 0);
+  var patternLength = pattern[0].length;
+  assert(patternLength > 0);
+  var sum = 0;
+  for (var hand in pattern) {
+    for (var time in pattern[hand]) {
+      sum += pattern[hand][time].throwNum;
+    }
+  }
+  var numBalls = sum / patternLength;
+  assert(numBalls == Math.round(numBalls));
+  assert(numBalls > 0);
+
+  var ballsToAllocate = numBalls;
+  var ballPaths = [];
+  // NOTE: The indices of locationsChecked are reversed from those of pattern
+  // for simplicity of allocation.  This might be worth flipping to match.
+  var locationsChecked = [];
+  for (var time = 0; time < patternLength && ballsToAllocate; ++time) {
+    locationsChecked[time] = locationsChecked[time] || [];
+    for (var hand = 0; hand < numHands && ballsToAllocate; ++hand) {
+      if (locationsChecked[time][hand]) {
+        continue;
+      }
+      var curThrowDesc = pattern[hand][time];
+      var curThrow = curThrowDesc.throwNum;
+      if (!curThrow) {
+        assert(curThrow === 0);
+        continue;
+      }
+      var throwDescriptors = [];
+      var curTime = time;
+      var curHand = hand;
+      var wraps = 0;
+      do {
+        if (!locationsChecked[curTime]) {
+          locationsChecked[curTime] = [];
+        }
+        assert(!locationsChecked[curTime][curHand]);
+        locationsChecked[curTime][curHand] = true;
+        // We copy curThrowDesc here, adding wraps * patternLength, to get
+        // the true throw time relative to offset.  Later we'll add in offset
+        // when we clone again, then mod by pathLength.
+        throwDescriptors.push(curThrowDesc.clone(wraps * patternLength));
+        var nextThrowTime = curThrow + curTime;
+        wraps += Math.floor(nextThrowTime / patternLength);
+        curTime = nextThrowTime % patternLength;
+        assert(curTime >= time); // Else we'd have covered it earlier.
+        curHand = curThrowDesc.destHand;
+        var tempThrowDesc = curThrowDesc;
+        curThrowDesc = pattern[curHand][curTime];
+        curThrow = curThrowDesc.throwNum;
+        assert(tempThrowDesc.destHand == curThrowDesc.sourceHand);
+        assert(curThrowDesc.time ==
+            (tempThrowDesc.throwNum + tempThrowDesc.time) % patternLength);
+      } while (curTime != time || curHand != hand);
+      var pathLength = wraps * patternLength;
+      var ballPath =
+        Path.ballPathFromThrowDescriptors(throwDescriptors, pathLength);
+      for (var i = 0; i < wraps; ++i) {
+        var offset = i * patternLength % pathLength;
+        ballPaths.push(ballPath.clone(offset, pathLength).fixUpModPathLength());
+      }
+      ballsToAllocate -= wraps;
+      assert(ballsToAllocate >= 0);
+    }
+  }
+  return ballPaths;
+}
+
+/**
+ * Given an array of ball paths, produce the corresponding set of hand paths.
+ * @param {!Array.Path} ballPaths the Paths of all the balls in the pattern.
+ * @param {!number} numHands how many hands to use in the pattern.
+ * @param {!number} patternLength the length, in beats, of the pattern.
+ * @return {!Array.Path} the paths of all the hands.
+ */
+function generateHandPaths(ballPaths, numHands, patternLength) {
+  assert(numHands > 0);
+  assert(patternLength > 0);
+  var handRecords = []; // One record per hand.
+  for (var idxBR in ballPaths) {
+    var descriptors = ballPaths[idxBR].descriptors;
+    for (var idxD in descriptors) {
+      var descriptor = descriptors[idxD];
+      // TODO: Fix likely needed for throws of 1.
+      if (!(descriptor instanceof ThrowDescriptor)) {
+        // It's a CarryDescriptor or a CarryOneDescriptor.
+        var hand = descriptor.hand;
+        if (!handRecords[hand]) {
+          handRecords[hand] = [];
+        }
+        // TODO: Should we not shorten stuff here if we're going to lengthen
+        // everything later anyway?  Is there a risk of inconsistency due to
+        // ball paths of different lengths?
+        var catchTime = descriptor.time % patternLength;
+        if (!handRecords[hand][catchTime]) {
+          // We pass in this offset to set the new descriptor's time to
+          // catchTime, so as to keep it within [0, patternLength).
+          handRecords[hand][catchTime] =
+              descriptor.clone(catchTime - descriptor.time);
+        } else {
+          assert(
+              handRecords[hand][catchTime].equalsWithMod(
+                  descriptor, patternLength));
+        }
+      }
+    }
+  }
+  var handPaths = [];
+  for (var hand in handRecords) {
+    var outDescriptors = [];
+    var inDescriptors = handRecords[hand];
+    var prevDescriptor = null;
+    var descriptor;
+    for (var idxD in inDescriptors) {
+      descriptor = inDescriptors[idxD];
+      assert(descriptor);  // Enumeration should skip array holes.
+      assert(descriptor.hand == hand);
+      if (prevDescriptor) {
+        outDescriptors.push(new EmptyHandDescriptor(prevDescriptor, descriptor,
+              patternLength));
+      }
+      outDescriptors.push(descriptor.clone());
+      prevDescriptor = descriptor;
+    }
+    // Note that this EmptyHandDescriptor that wraps around the end lives at the
+    // end of the array, not the beginning, despite the fact that it may be the
+    // active one at time zero.  This is the same behavior as with Paths for
+    // balls.
+    descriptor = new EmptyHandDescriptor(prevDescriptor, outDescriptors[0],
+        patternLength);
+    if (descriptor.time < outDescriptors[0].time) {
+      assert(descriptor.time + descriptor.duration == outDescriptors[0].time);
+      outDescriptors.unshift(descriptor);
+    } else {
+      assert(descriptor.time ==
+          outDescriptors[outDescriptors.length - 1].time + 1);
+      outDescriptors.push(descriptor);
+    }
+    handPaths[hand] =
+        new Path(outDescriptors, patternLength).fixUpModPathLength();
+  }
+  return handPaths;
+}
+
+// NOTE: All this Vector stuff does lots of object allocations.  If that's a
+// problem for your browser [e.g. IE6], you'd better stick with the embedded V8.
+// This code predates the creation of o3djs/math.js; I should probably switch it
+// over at some point, but for now it's not worth the trouble.
+
+/**
+ * A simple 3-dimensional vector.
+ * @constructor
+ */
+Vector = function(x, y, z) {
+  this.x = x;
+  this.y = y;
+  this.z = z;
+}
+
+Vector.prototype.sub = function(v) {
+  return new Vector(this.x - v.x, this.y - v.y, this.z - v.z);
+};
+
+Vector.prototype.add = function(v) {
+  return new Vector(this.x + v.x, this.y + v.y, this.z + v.z);
+};
+
+Vector.prototype.dot = function(v) {
+  return this.x * v.x + this.y * v.y + this.z * v.z;
+};
+
+Vector.prototype.length = function() {
+  return Math.sqrt(this.dot(this));
+};
+
+Vector.prototype.scale = function(s) {
+  return new Vector(this.x * s, this.y * s, this.z * s);
+};
+
+Vector.prototype.set = function(v) {
+  this.x = v.x;
+  this.y = v.y;
+  this.z = v.z;
+};
+
+Vector.prototype.normalize = function() {
+  var length = this.length();
+  assert(length);
+  this.set(this.scale(1 / length));
+  return this;
+};
+
+/**
+ * Convert the Vector to a string for debugging.
+ * @return {String} debugging output.
+ */
+Vector.prototype.toString = function() {
+  return '{' + this.x.toFixed(3) + ', ' + this.y.toFixed(3) + ', ' +
+      this.z.toFixed(3) + '}';
+};
+
+/**
+ * A container class that holds the positions relevant to a hand: where it is
+ * when it's not doing anything, where it likes to catch balls, and where it
+ * likes to throw balls to each of the other hands.
+ * @param {!Vector} basePosition the centroid of throw and catch positions when
+ * the hand throws to itself.
+ * @param {!Vector} catchPosition where the hand likes to catch balls.
+ * @constructor
+ */
+HandPositionRecord = function(basePosition, catchPosition) {
+  this.basePosition = basePosition;
+  this.catchPosition = catchPosition;
+  this.throwPositions = [];
+}
+
+/**
+ * Convert the HandPositionRecord to a string for debugging.
+ * @return {String} debugging output.
+ */
+HandPositionRecord.prototype.toString = function() {
+  var s = 'base: ' + this.basePosition.toString() + ';\n';
+  s += 'catch: ' + this.catchPosition.toString() + ';\n';
+  s += 'throws:\n';
+  for (var i = 0; i < this.throwPositions.length; ++i) {
+    s += '[' + i + '] ' + this.throwPositions[i].toString() + '\n';
+  }
+  return s;
+};
+
+/**
+ * Compute all the hand positions used in a pattern given a number of hands and
+ * a grouping style ["even" for evenly-spaced hands, "pairs" to group them in
+ * pairs, as with 2-handed jugglers].
+ * @param {!number} numHands the number of hands to use.
+ * @param {!String} style the grouping style.
+ * @return {!Array.HandPositionRecord} a full set of hand positions.
+ */
+function computeHandPositions(numHands, style) {
+  assert(numHands > 0);
+  var majorRadiusScale = 0.75;
+  var majorRadius = majorRadiusScale * (numHands - 1);
+  var throwCatchOffset = 0.45;
+  var catchRadius = majorRadius + throwCatchOffset;
+  var handPositionRecords = [];
+  for (var hand = 0; hand < numHands; ++hand) {
+    var circleFraction;
+    if (style == 'even') {
+      circleFraction = hand / numHands;
+    } else {
+      assert(style == 'pairs');
+      circleFraction = (hand + Math.floor(hand / 2)) / (1.5 * numHands);
+    }
+    var cos = Math.cos(Math.PI * 2 * circleFraction);
+    var sin = Math.sin(Math.PI * 2 * circleFraction);
+    var cX = catchRadius * cos;
+    var cY = 0;
+    var cZ = catchRadius * sin;
+    var bX = majorRadius * cos;
+    var bY = 0;
+    var bZ = majorRadius * sin;
+    handPositionRecords[hand] = new HandPositionRecord(
+      new Vector(bX, bY, bZ), new Vector(cX, cY, cZ));
+  }
+  // Now that we've got all the hands' base and catch positions, we need to
+  // compute the appropriate throw positions for each hand pair.
+  for (var source = 0; source < numHands; ++source) {
+    var throwHand = handPositionRecords[source];
+    for (var target = 0; target < numHands; ++target) {
+      var catchHand = handPositionRecords[target];
+      if (throwHand == catchHand) {
+        var baseV = throwHand.basePosition;
+        throwHand.throwPositions[target] =
+            baseV.add(baseV.sub(throwHand.catchPosition));
+      } else {
+        var directionV =
+            catchHand.catchPosition.sub(throwHand.basePosition).normalize();
+        var offsetV = directionV.scale(throwCatchOffset);
+        throwHand.throwPositions[target] =
+            throwHand.basePosition.add(offsetV);
+      }
+    }
+  }
+  return handPositionRecords;
+}
+
+/**
+ * Convert an array of HandPositionRecord to a string for debugging.
+ * @param {!Array.HandPositionRecord} positions the positions to display.
+ * @return {String} debugging output.
+ */
+function getStringFromHandPositions(positions) {
+  var s = '';
+  for (index in positions) {
+    s += positions[index].toString();
+  }
+  return s;
+}
+
+/**
+ * The set of curves an object passes through throughout a full animation cycle.
+ * @param {!number} duration the length of the animation in beats.
+ * @param {!Array.Curve} curves the full set of Curves.
+ * @constructor
+ */
+CurveSet = function(duration, curves) {
+  this.duration = duration;
+  this.curves = curves;
+}
+
+/**
+ * Looks up what curve is active at a particular time.  This is slower than
+ * getCurveForTime, but can be used even if no Curve starts precisely at
+ * unsafeTime % this.duration.
+ * @param {!number} unsafeTime the time at which to check.
+ * @return {!Curve} the curve active at unsafeTime.
+ */
+CurveSet.prototype.getCurveForUnsafeTime = function(unsafeTime) {
+  unsafeTime %= this.duration;
+  time = Math.floor(unsafeTime);
+  if (this.curves[time]) {
+    return this.curves[time];
+  }
+  var curve;
+  for (var i = time; i >= 0; --i) {
+    curve = this.curves[i];
+    if (curve) {
+      assert(i + curve.duration >= unsafeTime);
+      return curve;
+    }
+  }
+  // We must want the last one.  There's always a last one, given how we
+  // construct the CurveSets; they're sparse, but the length gets set by adding
+  // elements at the end.
+  curve = this.curves[this.curves.length - 1];
+  unsafeTime += this.duration;
+  assert(curve.startTime <= unsafeTime);
+  assert(curve.startTime + curve.duration > unsafeTime);
+  return curve;
+};
+
+/**
+ * Looks up what curve is active at a particular time.  This is faster than
+ * getCurveForUnsafeTime, but can only be used if if a Curve starts precisely at
+ * unsafeTime % this.duration.
+ * @param {!number} time the time at which to check.
+ * @return {!Curve} the curve starting at time.
+ */
+CurveSet.prototype.getCurveForTime = function(time) {
+  return this.curves[time % this.duration];
+};
+
+/**
+ * Convert the CurveSet to a string for debugging.
+ * @return {String} debugging output.
+ */
+CurveSet.prototype.toString = function() {
+  var s = 'Duration: ' + this.duration + '\n';
+  for (var c in this.curves) {
+    s += this.curves[c].toString();
+  }
+  return s;
+};
+
+/**
+ * Namespace object to hold the pure math functions.
+ * TODO: Consider just rolling these into the Pattern object, when it gets
+ * created.
+ */
+var JugglingMath = {};
+
+/**
+ * Computes the greatest common devisor of integers a and b.
+ * @param {!number} a an integer.
+ * @param {!number} b an integer.
+ * @return {!number} the GCD of a and b.
+ */
+JugglingMath.computeGCD = function(a, b) {
+  assert(Math.round(a) == a);
+  assert(Math.round(b) == b);
+  assert(a >= 0);
+  assert(b >= 0);
+  if (!b) {
+    return a;
+  } else {
+    return JugglingMath.computeGCD(b, a % b);
+  }
+}
+
+/**
+ * Computes the least common multiple of integers a and b, by making use of the
+ * fact that LCM(a, b) * GCD(a, b) == a * b.
+ * @param {!number} a an integer.
+ * @param {!number} b an integer.
+ * @return {!number} the LCM of a and b.
+ */
+JugglingMath.computeLCM = function(a, b) {
+  assert(Math.round(a) == a);
+  assert(Math.round(b) == b);
+  assert(a >= 0);
+  assert(b >= 0);
+  var ret = a * b / JugglingMath.computeGCD(a, b);
+  assert(Math.round(ret) == ret);
+  return ret;
+}
+
+/**
+ * Given a Path and a set of hand positions, compute the corresponding set of
+ * Curves.
+ * @param {!Path} path the path of an object.
+ * @param {!Array.HandPositionRecord} handPositions the positions of the hands
+ * juggling the pattern containing the path.
+ * @return {!CurveSet} the full set of curves.
+ */
+CurveSet.getCurveSetFromPath = function(path, handPositions) {
+  var curves = [];
+  var pathLength = path.pathLength;
+  for (var index in path.descriptors) {
+    var descriptor = path.descriptors[index];
+    var curve = descriptor.generateCurve(handPositions);
+    assert(!curves[curve.startTime]);
+    assert(curve.startTime < pathLength);
+    curves[curve.startTime] = curve;
+  }
+  return new CurveSet(pathLength, curves);
+}
+
+/**
+ * Given a set of Paths and a set of hand positions, compute the corresponding
+ * CurveSets.
+ * @param {!Array.Path} paths the paths of a number of objects.
+ * @param {!Array.HandPositionRecord} handPositions the positions of the hands
+ * juggling the pattern containing the paths.
+ * @return {!Array.CurveSet} the CurveSets.
+ */
+CurveSet.getCurveSetsFromPaths = function(paths, handPositions) {
+  var curveSets = [];
+  for (var index in paths) {
+    var path = paths[index];
+    curveSets[index] = CurveSet.getCurveSetFromPath(path, handPositions);
+  }
+  return curveSets;
+}
+
+/**
+ * This is a temporary top-level calculation function that converts a standard
+ * asynchronous siteswap, expressed as a string of digits, into a full
+ * ready-to-animate set of CurveSets.  Later on we'll be using an interface that
+ * can create a richer set of patterns than those expressable in the traditional
+ * string-of-ints format.
+ * @param {!String} patternString the siteswap.
+ * @param {!number} numHands the number of hands to use for the pattern.
+ * @param {!String} style how to space the hands ["pairs" or "even"].
+ * @return {!Object} a fully-analyzed pattern as CurveSets and associated data.
+ */
+function computeFullPatternFromString(patternString, numHands, style) {
+  var patternAsStrings = patternString.split(/[ ,]+ */);
+  var patternSegment = [];
+  for (var index in patternAsStrings) {
+    if (patternAsStrings[index]) {  // Beware extra whitespace at the ends.
+      patternSegment.push(parseInt(patternAsStrings[index]));
+    }
+  }
+  var pattern = [];
+  // Now expand the pattern out to the length of the LCM of pattern length and
+  // number of hands, so that each throw gets done in each of its incarnations.
+  var multiple = JugglingMath.computeLCM(patternSegment.length, numHands) /
+      patternSegment.length;
+  for (var i = 0; i < multiple; ++i) {
+    pattern = pattern.concat(patternSegment);
+  }
+
+  var fullPattern = expandPattern(pattern, numHands);
+  var patternLength = fullPattern[0].length;
+
+  var ballPaths = generateBallPaths(fullPattern);
+  var handPaths = generateHandPaths(ballPaths, numHands, patternLength);
+
+  var handPositions = computeHandPositions(numHands, style);
+  var ballCurveSets = CurveSet.getCurveSetsFromPaths(ballPaths, handPositions);
+  var handCurveSets = CurveSet.getCurveSetsFromPaths(handPaths, handPositions);
+
+  // Find the LCM of all the curveSet durations.  This will be the length of the
+  // fully-expanded queue.  We could expand to this before computing the
+  // CurveSets, but this way's probably just a little cheaper.
+  var lcmDuration = 1;
+  for (var i in ballCurveSets) {
+    var duration = ballCurveSets[i].duration;
+    if (duration > lcmDuration || lcmDuration % duration) {
+      lcmDuration = JugglingMath.computeLCM(lcmDuration, duration);
+    }
+  }
+  for (var i in handCurveSets) {
+    var duration = handCurveSets[i].duration;
+    if (duration > lcmDuration || lcmDuration % duration) {
+      lcmDuration = JugglingMath.computeLCM(lcmDuration, duration);
+    }
+  }
+  return {
+    numBalls: ballPaths.length,
+    numHands: handPaths.length,
+    duration: lcmDuration,
+    handCurveSets: handCurveSets,
+    ballCurveSets: ballCurveSets
+  }
+}