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