How to draw an ellipse ?

Physically :

If you have a loop of string and two nails at the focii, you can quickly sketch an ellipse

Algebraically

You need the semi major axis, a , and the semi minor axis , b, and a form of the ellipse to calculate some in between points :

so

If x^2 /a^2 + y^2 /b^2 = 1

a and b are needed for the intercepts

and y = ± b/a√(a^2-x^2) will give in between points.

If our ellipse was x^2/36 +y^2/25 = 1

then we get the four points ( 6,0) (-6,0) ( 0,5) and ( 0,-5) from a^2 = 36 and b^2 =25

Then use y = ±(5/6)√(36-x^2) and a calculator to get points between 0 and 6

.

Once you have enough points in Quadrant I , you can copy all those points via reflection to the other three quadrants , and draw a smooth curve thru all the points .

Draw a circle first for the skeleton

Then stretch the radius (equally distance though) as much as you like

if you have a ti-84 and higher it can do it