What is the equation of a circle with radius r and centre (a,b)
(x-a)^2 + (y-b)^2 = r^2
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Conic section
Shapes that are obtained by taking different plane slices through a __double cone__
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__Menaechmus__
(Greek mathematician) discovered conic sections around __350 BC__
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Ellipse
* An area where the sum of the distances from 2 fixed points on the plane remain constant
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Non-degenerate conics
* __parabolas, ellipses, or hyperbolas__
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Degenerate conics
* A single point, line, and pair of lines
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Circle
* a __type of conic,__ a set of points that lie at a fixed distance (radius), from a fixed point (centre)
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Orthogonal
Two intersecting circles that meet at __right angles__
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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
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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.
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**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.__
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**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**)
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**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
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* **Auxiliary circle**-
A circle whose diameter is its major axis
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Standard form of conics
the centre is at the __origin__, and the axes are parallel to the __x and y axis__
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Matrices equation for conics
**x^T Ax + J^T x + H = 0.**
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Quadric surfaces
* surfaces in R^3 that are the __natural analogues__ of the conics.
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Degenerate Quadrics
* Empty set, single point, single line, single plane, pair of planes, and a cylinder
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Cyllinder
* any surface that consists of an ellipse, parabola, or hyperbola in some plane
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Ruled surface in R^3
a surface that can be made up from a family of __straight lines__
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Family of generators
* straight lines that __construct__ the hyperboloid
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