In many textbooks, the two radii are specified as being the Click "show details" to check your answers.Write the equations of the ellipse in parametric form.Drag the five orange dots to create a new ellipse at a new center point.In the above applet click 'reset', and 'hide details'.In fact, all the circles and ellipses in the applets on this site are drawn using this equation form.įor more on this see An Algorithm for Drawing Circles. This form of defining an ellipse is very useful in computer algorithms that draw circles and ellipses. Using the Pythagorean Theorem to find the points on the ellipse, we get the more common form of the equation.įor more see General equation of an ellipse Algorithm for drawing ellipses However, when you graph the ellipse using the parametric equations, simply allow t to range from 0 to 2π radians to find the (x, y) coordinates for each value of t.
So as you can see, the angle t is not the same as the angle that the point on the ellipse subtends at the center. This causes the ellipse to be wider than the circle by a factor of two, whereas the height remains the same, as directed by the values 2 and 1 in the ellipse's equations. So to get the corresponding point on the ellipse, the x coordinate is multiplied by two, thus moving it to the right. The angle t defines a point on the circle which has the coordinates In the figure below we start with a circle, and for simplicity give it a radius of one To see why this is so, consider an ellipse as a circle that has been stretched or squashed along each axis. The parameter t can be a little confusing with ellipses.įor any value of t, there will be a corresponding point on the ellipse.īut t is not the angle subtended by that point at the center. This demonstrates that a circle is just a special case of an ellipse. This is a circle, and the equations for it look just like the
In the applet above, drag the right orange dot left until the two radii are the same. These are the same equations as for a circle.
In the applet above, drag the orange dot at the center to move the ellipse, and note how the equations change to match.Īlso, adjust the ellipse so that a and b are the same length, and convince yourself that in this case, (h,k) are the x and y coordinates of the ellipse's center. A is the radius along the x-axis ( * See radii note below )
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