Equation of an Ellipse in JavaScript - Draw and Animate an Elliptical Path
Understanding the Ellipse Equation | Maths Explanation for JavaScript Kids
In geometry, an ellipse can be defined as the set of all points where the sum of the distances to two foci is constant.
The general form of an ellipse equation is (x - h)² / a² + (y - k)² / b² = 1. We'll use this to calculate ellipse points dynamically in JavaScript.
In this tutorial, we'll explore the ellipse equation in JavaScript, derive its conic form, and show how to draw and animate ellipses on an HTML5 canvas using simple JavaScript code.
Using the Equation of an Ellipse in JavaScript
Ellipses are represented by the general equation
(x - h)2
+
(y - k)2
= 1
a2
b2
where (h, k) is the centre (point of intersection of the major and minor axes) of the ellipse; and
a & b are the major and minor radii (1/2 the axis) respectively.
Figure: Ellipse equation diagram in JavaScript showing major and minor axes
To animate motion along the ellipse in JavaScript, we'll express *y* in terms of *x*.
Solving for y, we have:
(y - k)2
= 1 -
(x - h)2
b2
a2
(y - k)2
=
a2 - (x - h)2
b2
a2
(y - k)2 = b2[
a2 - (x - h)2
]
a2
(y - k) =
±b√(a2 - (x - h)2)
a
y = k ± b/a√(a2 - (x - h)2)
This gives us two values for *y* (the upper and lower halves of the ellipse).
We can use these values to plot and animate a moving object in JavaScript on both curves.
Parametric Equations of a Ellipse | Maths Explanation for JavaScript Kids
Elliptical motion can also be represented parametrically:
x = h + a * cos(θ)
y = k + b * sin(θ)
These equations allow you to move an object smoothly along an elliptical trajectory on the canvas in JavaScript,
especially for animations, games, and physics simulations.
Animating Elliptical Motion with JavaScript
To move a small image body (or dot) along an elliptical trajectory, we'll increment *x* from *(h - a)* to *(h + a)*
and compute *y* using the ellipse equation in JavaScript.
Create a new file; On Notepad++: File, New.
Call it elliptical_path.html.
Type out the adjoining JavaScript code for animating an image body through the path of an ellipse.
How the JavaScript Ellipse Equation Animation Code Works
When you click the Move button, a dot will trace the upper and lower halves of the ellipse,
showing how the ellipse equation in JavaScript generates smooth, curved motion.
Key Takeaways on Elliptical Path Animation in JavaScript
In this tutorial, you've learned that:
The general form of the ellipse equation is (x - h)² / a² + (y - k)² / b² = 1
With an ellipse equation, you can draw and animate an ellipse in JavaScript
Applications of elliptical path animation include use in graphics, game design, and physics simulations
Applications of Ellipse Equations
Elliptical motion is used in physics simulations, orbital paths, and data visualization.
By understanding the ellipse equation in JavaScript, you can create animations,
model celestial orbits, or generate smooth transitions in web graphics.
FAQs: Ellipse Equation and JavaScript
What is the equation of an ellipse in JavaScript?
The equation (x - h)² / a² + (y - k)² / b² = 1 can be implemented using JavaScript
to compute ellipse points or draw an ellipse on a canvas element.
How else do you animate an ellipse path on canvas?
Use the parametric form x = h + a * cos(t), y = k + b * sin(t), and increment θ over time using
requestAnimationFrame() to create smooth elliptical motion.
Can you draw ellipses using the Canvas API directly?
Yes. The ctx.ellipse() method in modern browsers allows you to draw an ellipse directly without manually computing each point.
Summary: Visualizing ellipse Equation in JavaScript
Ellipses are fundamental conic sections, and in JavaScript, you can use their equations to draw and
animate motion along an elliptical path on an HTML5 canvas.
In this lesson, we've derived the ellipse equation, express it in code,
and use JavaScript to animate an image body moving through an ellipse.
So! JavaScript Fun Practice Exercise - Animate along Elliptical Path
As a fun practice exercise, try adjusting the foci points - a, b;
and the major and minor radii - h, k; to change the ellipse's position and size.
This will be a great way to connect mathematics and programming, and help you
understand more about JavaScript animations and ellipse equations.
JavaScript Animation Code for elliptical_path.html
<canvasid="move_ellip"width="500"height="350"style="border: 1px solid #000000;">
Your browser (version) does not support canvas object; Time to update! </canvas> <buttononclick="moveElliptic()">Move</button>
<script> var canvas = document.getElementById("move_ellip"); var context = canvas.getContext("2d");
context.fillStyle ="#888888"; // color for our body(circle)
var h =250; var k =175; var a =150; var b =100; var x = h - a; var y = k; // draw a dot
context.beginPath();
context.arc(x, y, 5, 0, 2*Math.PI);
context.fill();
function moveElliptic(){ // condition for continuing motion if(x <= h + a){
y = k -(b / a)*Math.sqrt(Math.pow(a, 2)-Math.pow((x - h), 2)); // redraw dot (upper curve)
context.beginPath();
context.arc(x, y, 5, 0, 2*Math.PI);
context.fill();
y = k +(b / a)*Math.sqrt(Math.pow(a, 2)-Math.pow((x - h), 2)); // redraw dot (lower curve)
context.beginPath();
context.arc(x, y, 5, 0, 2*Math.PI);
context.fill();
x +=20;
setTimeout(function(){ moveElliptic(); }, 200); } } </script>
