The normal at (x1, y1) is a^2x/x1 - b^2y/y1 = a^2 - b^2. In parametric form: ax/cos(theta) - by/sin(theta) = a^2 - b^2. In slope form: y = mx - m(a^2-b^2)/sqrt(a^2 + b^2m^2). From an external point, at most 4 normals can be drawn (contrast: 3 for parabola). The normal at P bisects the angle between the focal radii SP and S'P (the internal angle), while the tangent bisects the external angle. This is the reflection property: a ray from one focus reflects through the other focus. If the normal at eccentric angle theta meets the major axis at G, then CG = e^2a*cos(theta). The evolute (envelope of normals) is the astroid (ax)^(2/3) + (by)^(2/3) = (a^2-b^2)^(2/3).