Chapter 1-Conics

(x-a)^2 + (y-b)^2 = r^2

Shapes that are obtained by taking different plane slices through a double cone

(Greek mathematician) discovered conic sections around 350 BC

• An area where the sum of the distances from 2 fixed points on the plane remain constant

• parabolas, ellipses, or hyperbolas

• A single point, line, and pair of lines

• a type of conic, a set of points that lie at a fixed distance (radius), from a fixed point (centre)

Two intersecting circles that meet at right angles

• It is an ellipse if e (eccentricity) is between 0 and 1, a parabola of e equals 1, and a hyperbola is e is greater than 1

The angle that incoming light makes with the tangent to a surface is the same as the angle that the reflected light makes with the tangent.

light which comes from one focus of an elliptical mirror is reflected at the ellipse to pass through the second focus.

light coming from one focus of a hyperbolic mirror is reflected at the hyperbola and makes the light appear to have come from the other focus (Internal Reflection property). Light going towards one focus is reflected towards the other focus (External Reflection Property)

incoming light parallel to the axis is reflected at the parabola to pass through the focus. Light coming from the focus of a parabola is reflected to give a beam of light parallel to the axis of the parabola

A circle whose diameter is its major axis

the centre is at the origin, and the axes are parallel to the x and y axis

x^T Ax + J^T x + H = 0.

• surfaces in R^3 that are the natural analogues of the conics.

• Empty set, single point, single line, single plane, pair of planes, and a cylinder

• any surface that consists of an ellipse, parabola, or hyperbola in some plane

a surface that can be made up from a family of straight lines

• straight lines that construct the hyperboloid