Chapter 1-Conics

What is the equation of a circle with radius r and centre (a,b)

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

Conic section

Shapes that are obtained by taking different plane slices through a __double cone__

__Menaechmus__

(Greek mathematician) discovered conic sections around __350 BC__

Ellipse

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

Non-degenerate conics

* __parabolas, ellipses, or hyperbolas__

Degenerate conics

* A single point, line, and pair of lines

Circle

* a __type of conic,__ a set of points that lie at a fixed distance (radius), from a fixed point (centre)

Orthogonal

Two intersecting circles that meet at __right angles__

Eccentricity

* 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

Reflection law

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.

**Reflection property of the Ellipse**

light which comes from one focus of an __elliptical mirror__ is reflected at the ellipse to pass through the __second focus.__

**Reflection property of the hyperbola-**

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**)

**Reflection Property of the Parabola**-

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

* **Auxiliary circle**-

A circle whose diameter is its major axis

Standard form of conics

the centre is at the __origin__, and the axes are parallel to the __x and y axis__

Matrices equation for conics

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

Quadric surfaces

* surfaces in R^3 that are the __natural analogues__ of the conics.

Degenerate Quadrics

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

Cyllinder

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

Ruled surface in R^3

a surface that can be made up from a family of __straight lines__

Family of generators

* straight lines that __construct__ the hyperboloid