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Alt 04-07-2009, 11:00   #1
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Standart Analytic Geometry in 3-Dimension

Analytic Geometry in 3-Dimension

A. Lines & Planes in R³

A plane in space can be specified by giving its inclination and specifying one of its points. Let us to write the equation of the plane passing through the point and having the nonzero vector as a normal. If we have another point , the vector is orthogonal to where can be written as
(1)
Since , we rewrite equation (1) we get

(2)
This is called point-normal form equation of the plane.

If are not all zero constans then
(3)
is a plane having the vector as a normal. Equation (3) is general form of the equation of a plane.
Now we shall write equation of line in space. Let be the line pass through point and parallel to nonzero vector . If we have point lying on , we can write equation (4) where is a scalar
(4)
In terms of components equation (4) can be written as
where (5)
These equations are called parametric equations for line through parallel to .
Also sometimes it is written instead of (5) as
(6)
where parameter is eliminated. These are called symmetric equations for the line.
The distance from a point to a line through parallel to a vector is
(7)




B. Quadric Surfaces in R³

A quadric surface is the graph in space of a second-degree equation (quatrtic) in .
The general quadratic form is

(8)
where are not all zero constans.
Also is called associated quadratic form.
Some of the quadric surfaces and their equations are;
Sphere:


Ellipsoid:



Elliptic Paraboloid:


Circular Paraboloid or Paraboloid of Revolution:


Elliptic Cone:


Hyperboloid of One Sheet:


Hyperboloid of One Sheets:


Hyperboloic Paraboloid:


Torus:


Superquadrics are generalization of quadratic representations. They include additional parmeters to provide flexibility of shapes of quadric equations. Two of them are;
Superellipse:


Superellipsoid:



References:
1.- THOMAS, G. B. and R. FINNEY (1998). Calculus and Analytic Geometry, Addison-Wesley, Reading, MA.
2.- ANTON, H. (1991). Elementary Linear Algebra, John Willey & Sons, New York
3.- HEARN, D. and M. P. BAKER (1994). Computer Graphics, Prentice Hall, Upper Saddle River, NJ.
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