focal distance and eccentricity defined

Definition of Focal Distance and Eccentricity for an Ellipse PhL 11.13.16 This document is a definition of the focal distance of an ellipse. 1. Start with this ellipse centered at the origin + = 1 which can be written as = 1 b2x2 + a2y2 - a2b2 = 0 You could solve this for y2 as follows: (b/a)2x2 + y2 - b2 = 0 y2 = b2 - (b/a)2x2 (1) 2. Consider the two points (±c,0) where c is at the moment just a number. Run a string from the left point (-c,0) to some (x,y) on the ellipse, and thence from (x,y) to the right point (c,0). Let the length of this string be 2L where in general L = L(x,y). The total length 2L of the string is this 2L(x,y) = + ≡ + so 4L2 = M + P + 2 4L2 - M - P = 2 [4L2 - M - P ]2 = 4MP f = [4L2 - M - P]2 - 4MP = 0 Maple simplifies this result so that f/16 is the following Therefore, L4 - (x2+y2+c2)L2 + x2c2 = 0 (2) L4 - (x2+[b2 - (b/a)2x2 ]+c2)L2 + x2c2 = 0 For some general value of c, this is a quadratic equation in L2 which one could solve and one would obtain some messy solution L(x; a,b,c). The length 2L is different for different points (x,y) on the ellipse. But now go back to (2) above and just process things again : L4 - x2L2-y2L2-c2L2 + x2c2 = 0 L4 - x2L2- [b2 - (b/a)2x2 ]L2-c2L2 + x2c2 = 0 // using (1) here L4 - x2L2 - b2L2 + (b/a)2L2x2- c2L2 + x2c2 = 0 a2L4 - x2a2L2 - a2b2L2 + b2L2x2- a2c2L2 + x2a2c2 = 0 x2 [ -a2L2 + b2L2 + a2c2 ] + [a2L4 - a2b2L2 - a2c2L2] = 0 or x2 [ -a2L2 + b2L2 + a2c2 ] + a2L2[L2 - b2 - c2] = 0 Question: Is there a value for c such that this equation is true for all x on the ellipse? In order for this equation to be true for any x, we must have these two equations be valid -a2L2 + b2L2 + a2c2 = 0 L2 - b2 - c2 = 0 . // L2 = b2+c2 The second says L2 = b2+c2 and then the first says -a2(b2+c2) + b2(b2+c2) + a2c2 = 0 -a2b2 - a2c2 + b4 + b2c2 + a2c2= 0 -a2b2 + b4 + b2c2 = 0 -a2 + b2 + c2 = 0 c2 = a2- b2 The conclusion is that if we set c = , then the distance 2L is the same for all x, and the distance is in fact 2L = 2a. This is the only value of c for which the string length will be constant. We have just proven this theorem: Theorem: Start with an ellipse with semi-axes a and b centered at the origin, a > b. Run a string from the left point (-c,0) to some (x,y) on the ellipse, and thence from (x,y) to the right point (c,0). Let the length of this string be 2L where in general L = L(x,y). Then: (1) if we set c = , then the length of the string is 2L(x,y) = 2a which is independent of (x,y). (2) If c is any other value, then the length of the string length is NOT constant. Definition: The points ±c which make the string length constant are called "the focal points" of the ellipse. Definition: The eccentricity of the ellipse is defined as ε = c/a. If c ≈ 0, the two focal points are close to the center point and the ellipse is close to a circle. If ε → 1, the ellipse is very wide and thin.