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Figure 5-12. Multiple balls of different sizes bouncing off one another with friction

applied

Curve and Circular Movement

Whew! Moving and colliding balls on vectors can create some cool effects. However,

moving in straight lines is not the only way you might want to move objects. In this

section, we will show you some ways to animate objects by using circles, spirals, and

curves.

Uniform Circular Motion

Uniform circular motion occurs when we move an object along the distinct radius of a

defined circle. When we know the radius, we can use our old friends cosine and sine

to find the x and y locations of the moving object. The equations to find the locations

of an object moving uniformly on a defined circle are as follows:

x = radius * cosine(angle)

y = radius * sine(angle)

We will create an example of uniform circular movement with a circle that has a radius

of 125, with its center position at 250,250 on the canvas. We will move a ball along that

circle, starting at an angle of 0.

Curve and Circular Movement | 239

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In canvasApp(), we will define this circle path as a dynamic object stored in the circle

variable. While this object defines the properties of a circle, we will not actually draw

this circle on the canvas; rather, it defines only the path on which we will move our ball

object:

var circle = {centerX:250, centerY:250, radius:125, angle:0}

var ball = {x:0, y:0,speed:.1};

In drawScreen(), we will incorporate the equations for uniform circular movement. To

do this, we will set the x and y properties of the ball object to the products of the

equations, added to the center location of the circle path on the canvas (circle.cen

terX, circle.centerY):

ball.x = circle.centerX + Math.cos(circle.angle) * circle.radius;

ball.y = circle.centerY + Math.sin(circle.angle) * circle.radius;

We then add the speed of the ball to the angle of the circle path. This effectively sets the

ball to move to a new location the next time drawScreen() is called:

circle.angle += ball.speed;

Finally, we draw the ball onto the canvas:

context.fillStyle = "#000000";

context.beginPath();

context.arc(ball.x,ball.y,15,0,Math.PI*2,true);

context.closePath();

context.fill();

You can see what the circle path looks like in Figure 5-13. We have drawn the points on

the canvas to illustrate the circle path.

You can easily alter the location and size of the circle path by altering the radius,

centerX, and centerY properties of the circle path object.

Example 5-9 shows the code for CH5EX9.html.

240 | Chapter 5: Math, Physics, and Animation

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Figure 5-13. Moving an object in a circle

Example 5-9. Moving in a circle

<!doctype html>

<head>

<title>CH5EX9: Moving In A Circle</title>

window.addEventListener('load', eventWindowLoaded, false);

function eventWindowLoaded() {

canvasApp();

}

function canvasSupport () {

return Modernizr.canvas;

}

function canvasApp() {

if (!canvasSupport()) {

return;

}

Curve and Circular Movement | 241

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function drawScreen () {

context.fillStyle = '#EEEEEE';

context.fillRect(0, 0, theCanvas.width, theCanvas.height);

//Box

context.strokeStyle = '#000000';

context.strokeRect(1, 1, theCanvas.width-2, theCanvas.height-2);

ball.x = circle.centerX + Math.cos(circle.angle) * circle.radius;

ball.y = circle.centerY + Math.sin(circle.angle) * circle.radius;

circle.angle += ball.speed;

context.fillStyle = "#000000";

context.beginPath();

context.arc(ball.x,ball.y,15,0,Math.PI*2,true);

context.closePath();

context.fill();

}

var radius = 100;

var circle = {centerX:250, centerY:250, radius:125, angle:0}

var ball = {x:0, y:0,speed:.1};

theCanvas = document.getElementById("canvasOne");

context = theCanvas.getContext("2d");

function gameLoop() {

window.setTimeout(gameLoop, 20);

drawScreen()

}

gameLoop();

}

</script>

</head>

<body>

Your browser does not support HTML5 Canvas.

</canvas>

</div>

</body>

</html>

242 | Chapter 5: Math, Physics, and Animation

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Moving in a Simple Spiral

There are many complicated ways to move an object on a spiral path. One such way

would be to use the Fibonacci sequence, which describes a pattern seen in nature that

appears to create perfect spirals. The Fibonacci sequence starts with the number 0, and

continues with each subsequent number calculated as the sum of the two previous

numbers in the sequence. Each subsequent rotation of the spiral is the sum of the two

previous numbers (1, 2, 3, 5, 8, 13, 21, 34, 55, 89...). However, as you might imagine, the

math used to create this sequence is quite involved, and it is also difficult to translate to

object movement.

For our purposes, we can create a simple spiral by increasing the radius of the circle

path on each call to drawScreen(). If we take the code from Example 5-9, we would add

a radiusInc variable, which we will use as the value to add the radius movement path

of the circle. We create this new variable in canvasApp():

var radiusInc = 2;

Then, in drawScreen(), we add the following code to increase the radius of the circle

every time we move the object:

circle.radius += radiusInc;

In Figure 5-14, you can see what the resulting spiral looks like. (To illustrate the path,

this example includes the points.)

If you want a tighter spiral, decrease the value of radiusInc. Conversely, if you want a

wider spiral, increase the value of radiusInc.

Example 5-10 shows the code for CH5EX10.html from the code distribution.

Figure 5-14. Moving an object in a simple spiral pattern

Curve and Circular Movement | 243

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Example 5-10. Moving in a simple geometric spiral

<!doctype html>

<head>

<title>CH5EX10: Moving In A Simple Geometric Spiral </title>

window.addEventListener('load', eventWindowLoaded, false);

function eventWindowLoaded() {

canvasApp();

}

function canvasSupport () {

return Modernizr.canvas;

}

function canvasApp() {

if (!canvasSupport()) {

return;

}

var pointImage = new Image();

pointImage.src = "point.png";

function drawScreen () {

context.fillStyle = '#EEEEEE';

context.fillRect(0, 0, theCanvas.width, theCanvas.height);

//Box

context.strokeStyle = '#000000';

context.strokeRect(1, 1, theCanvas.width-2, theCanvas.height-2);

ball.x = circle.centerX + Math.cos(circle.angle) * circle.radius;

ball.y = circle.centerY + Math.sin(circle.angle) * circle.radius;

circle.angle += ball.speed;

circle.radius += radiusInc;

//Draw points to illustrate path

points.push({x:ball.x,y:ball.y});

for (var i = 0; i< points.length; i++) {

context.drawImage(pointImage, points[i].x, points[i].y,1,1);

}

context.fillStyle = "#000000";

244 | Chapter 5: Math, Physics, and Animation

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context.beginPath();

context.arc(ball.x,ball.y,15,0,Math.PI*2,true);

context.closePath();

context.fill();

}

var radiusInc = 2;

var circle = {centerX:250, centerY:250, radius:2, angle:0, radiusInc:2}

var ball = {x:0, y:0,speed:.1};

var points = new Array();

theCanvas = document.getElementById("canvasOne");

context = theCanvas.getContext("2d");

function gameLoop() {

window.setTimeout(gameLoop, 20);

drawScreen()

}

gameLoop();

}

</script>

</head>

<body>

Your browser does not support HTML5 Canvas.

</canvas>

</div>

</body>

</html>

Cubic Bezier Curve Movement

Cubic Bezier curves can be used to define a movement path for an object. Pierre Bezier

first popularized these curves in the 1960s. They are widely used in 2D vector graphics

to define smooth curves for drawing, but they can also be used in animation to define

a path for motion.

A cubic Bezier curve is created using four distinct points—p0, p1, p2, and p3:

The starting point of the curve. We will refer to these x and y values as x0 and y0.

The ending point of the curve. We will refer to these x and y values as x3 and y3.

Curve and Circular Movement | 245

p1 and p2

The control points for the curve. The curve does not pass through these points;

instead, the equation uses these points to determine the arc of the curve. We will

refer to these x and y values as x0, x1, x2, x3, y0, y1, y2, and y3.

The usage of the p1 and p2 points is the biggest stumbling block for

understanding Bezier curves. The easiest way to understand the rela‐

tionship between these points and the curve is to draw them on a bit‐

mapped canvas, which we will do several times in this chapter.

After you have the four points, you need to calculate six coefficient values that you will

use to find the x and y locations as you move an object on the curve. These coefficients

are known as ax, bx, cx, ay, by, and cy. They are calculated as follows:

cx = 3 * (x1 - x0)

bx = 3 *(x2 - x1) - cx

ax = x3 - x0 - cx - bx

cy = 3 * (y1 - y0)

by = 3 * (y2 - y1) - cy

ay = y3 - y0 - cy - by

After you’ve calculated the six coefficients, you can find the x and y locations based on

the changing t value using the following equations. The t value represents movement

over time:

x(t) = axt3 + bxt2 + cxt + x0

y(t) = ayt3 + byt2 + cyt + y0

For our purposes, the t value will be increased by the speed at which we want the object

to move. However, you will notice that this value does not easily equate to the speed

values we used elsewhere in this chapter. The reason is that the t value was not created

with movement over time for animation in mind. The speed we specify must be smaller

than 1 so that the movement on the curve will be incremental enough for us to see it as

part of the animation. For our example, we will increase t by a speed of .01 so that we

will see 100 points on the movement curve (1/100 = .01). This is advantageous because

we will know our object has finished moving when the t value is equal to 1.

For Example 5-11 (CH5EX11.html), we will start by creating the four points of the Bezier

curve in the canvasApp() function:

var p0 = {x:60, y:10};

var p1 = {x:70, y:200};

var p2 = {x:125, y:295};

var p3 = {x:350, y:350};

We then create a new ball object with a couple differing properties from those in the

other examples in this chapter. The speed is .01, which means that the object will move

246 | Chapter 5: Math, Physics, and Animation

100 points along the curve before it is finished. We start the t value at 0, which means

that the ball will begin at p0:

var ball = {x:0, y:0, speed:.01, t:0};

Next, in the drawScreen() function, we calculate the Bezier curve coefficient values (ax,

bx, cx, ay, by, cy) based on the four points (p0, p1, p2, p3):

var cx = 3 * (p1.x - p0.x);

var bx = 3 * (p2.x - p1.x) - cx;

var ax = p3.x - p0.x - cx - bx;

var cy = 3 * (p1.y - p0.y);

var by = 3 * (p2.y - p1.y) - cy;

var ay = p3.y - p0.y - cy - by;

Then we take our t value and use it with the coefficients to calculate the x and y values

for the moving object. First, we get the t value from the ball object, and we store it

locally so that we can use it in our calculations:

var t = ball.t;

Next we add the speed to the t value so that we can calculate the next point on the Bezier

path:

ball.t += ball.speed;

Then we use the t value to calculate the x and y values (xt, yt) using the Bezier curve

equations:

var xt = ax*(t*t*t) + bx*(t*t) + cx*t + p0.x;

var yt = ay*(t*t*t) + by*(t*t) + cy*t + p0.y;

We add the speed to the t value of ball and then check to see whether t is greater than

1. If so, we don’t increase it any further because we have finished moving on the curve:

ball.t += ball.speed;

if (ball.t > 1) {

ball.t = 1;

}

Finally, when we draw the ball object on the canvas, we use the xt and yt values:

context.arc(xt,yt,5,0,Math.PI*2,true);

Figure 5-15 shows what Example 5-11 (CH5EX11.html) looks like when it is executed

in a web browser. In addition to drawing the points of the path using the points array,

we also draw the four points of the Bezier curve. These illustrate the relationship of the

points to the curve itself. Notice that the curve does not pass through p1 or p2.

Curve and Circular Movement | 247

Figure 5-15. Moving a circle on a Bezier curve

Example 5-11 gives the full code listing for CH5EX11.html, including the code to draw

the Bezier curve points on the canvas. You can find that code in the drawScreen()

function following the //draw the points comment.

Example 5-11. Moving on a cubic Bezier curve

<!doctype html>

<head>

<title>CH5EX11: Moving On A Cubic Bezier Curve </title>

window.addEventListener('load', eventWindowLoaded, false);

function eventWindowLoaded() {

canvasApp();

}

function canvasSupport () {

248 | Chapter 5: Math, Physics, and Animation

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