What is the parametric equation of an ellipse?

Question

Christopher Agresta · Accepted Answer

Here is one example...

You can have #(nsin(t),mcos(t))# when #n!=m#, and #n# and #m# do not equal to #1#. This is essentially because: #=>x=nsin(t)# #=>x^2=n^2sin^2(t)# #=>x^2/n^2=sin^2(t)# #=>y=mcos(t)# #=>y^2/m^2=cos^2(t)# #=>x^2/n^2+y^2/m^2=sin^2(t)+cos^2(t)# Using the fact that #sin^2(x)+cos^2(x)=1#... #=>x^2/n^2+y^2/m^2=1# This is essentially an ellipse! Note that if you want a non-circle ellipse, you have to make sure that #n!=m#

Charles Arocha · Answer

undefined The parametric equations of an ellipse are given by:

x(t) = a * cos(t)
y(t) = b * sin(t)

Where:
- 'a' is the length of the semi-major axis (half of the longest diameter).
- 'b' is the length of the semi-minor axis (half of the shortest diameter).
- 't' is the parameter that varies as the point moves along the ellipse.
- 'cos' and 'sin' are the cosine and sine trigonometric functions, respectively.

These parametric equations trace out the ellipse as 't' varies from 0 to 2π radians (or 0 to 360 degrees), covering one complete revolution around the ellipse.