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How to Rotate an Ellipse

The ellipse is one of the many fundamental shapes found geometry, defined as a two-dimensional object with unequal vertical and horizontal axes. Although similar visually to a circle, the difference is clear when observing the radial properties -- the circle has a constant radius between any two points, whereas the ellipse will always be variable. Despite the increased complexity in comparison to the circle, the ellipse is more commonly used in mathematical and scientific methods as it produces a more realistic model, discovered by Joseph Kepler in the 17th century when charting planetary orbits.

Instructions

- 1
  Construct the ellipse you are about to measure by figuring out the coordinate points the ellipse lies on. As the general form of an ellipse is x^2 + y^2 = 1, by trying different values of x and rearranging the equation using a little algebraic knowledge, you will end up with a series of y values. Put the x and y values together in the form (x,y), resulting in a number of points that you can sketch a basic ellipse with. Remember that as you are dealing with squared numbers for every x value you place into the equation, you will get two y values back. For example, use the equation x^2 + (y^2)/2 = 1 and substituting {-1,0,1} for x produces coordinates of [-1,0], [0,√2], [0,-√2], [1,0]; sketching these points will confirm that the equation does produce an ellipse.
- 2
  Use the equations u = [x * cos(t) + y * sin(t)], v = [y * cos(t) - x * sin(t)]; where t is the angle of translation and (u,v) is the new rotated coordinate taking your values of (x,y) in the first step. Repeat this for all the points calculated in Step 1. Continuing the example from Step 1 and using t = 45 degrees, converting the (x,y) points using the u,v equations produces the series of points [-pi/2,0], [0, pi/√2], [0, -pi/√2], [0, pi/2].
- 3
  Plot the series of new (u,v) coordinates from Step 2 to produce a sketch of the translated ellipse, preferably on the sketch produced in Step 1, allowing you to visually estimate if the translation was successful. If you need to produce a line equation for the translation, solve by taking linear equations of the (u,v) equations about two points of the translated ellipse. Sketching the example will confirm a rotation of 90 degrees.

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