Conic sections

Cross-section of slices thru a double-napped (right circular) cone.


set of points whose distance r to center are the same.
Circle centered at origin (0,0):
Circle centered at (h,k):

Area of a circle = πr2. Circumference = 2πr.


set of points whose sum of distances d1+d2 to two foci F1,F2 are the same.
d1+d2 = 2a

Ellipses centered at origin (0,0):

Ellipses centered at (h,k):

Ellipses rotated:

Area of an ellipse = πab.
There is no algebraic formula for the circumference. But various approximations to within .0015% accuracy exist. Simple: C≈2π((a+b)/2)
Ramanujan: C≈π[3(a+b)-√((3a+b)(a+3b))]
Eccentricity e is a measure of the ovalness/roundness of the ellipse. 0<e<1. Closer to 1: flatter.
Circle is an ellipse in which a=b, the eccentricity e is 0, and the two foci are the same point, the center of the circle.
Java applet to draw ellipse.


set of points that are the same distance from a point, the focus, and a line, the directrix.
==locus of points equi-distant from the focus and the directrix.

Each focus-directrix pair uniquely defines a parabola.
Vertical parabola is the graph of a quadratic function.
Given a horizontal directrix y=d and a focus (xf,yf)
the coefficients a, b, and c of the quadratic polynomial function are:
a= 1 / 2(yf-d)
b= xf / yf-d
c= xf2+yf2-d2 / 2(yf-d)

The eccentricity e of every parabola is 1.


set of points whose difference of distances d1-d2 to two foci F1,F2 are the same.
Hyperbolas centered at origin (0,0): f=√(a2+b2)    e=f/a
"Unit hyperbola":

Hyperbola centered at (h,k):

Distance from center to focus F is c=√(a2+b2)
The eccentricity e=c/a, e>1.

Conic section e Equation
Circle 0 general: Ax2+Cy2+Dx+Ey+F=0, A=C
center at origin: Ax2+Cy2+F=0, A=C
Ellipse 0<e<1 general: Ax2+Bxy+Cy2+Dx+Ey+F=0, B2<4AC
orthogonal,center at origin: Ax2+Cy2+F=0, A≠C
orthogonal,center not at origin: Ax2+Cy2+Dx+Ey+F=0, A≠C
Parabola 1 general: Ax2+Bxy+Cy2+Dx+Ey+F=0, B2=4AC
vertical: Ax2+Dx+Ey+F=0
horizontal: Cy2+Dx+Ey+F=0
Hyperbola >1 general: Ax2+Bxy+Cy2+Dx+Ey+F=0, B2>4AC
vertical, center at origin: -Ax2+Cy2+F=0
horizontal, center at origin: Ax2-Cy2+F=0