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Chapter 10:
Mathematical Formulation
Abstract
To create aspherical surfaces, a common approach uses polynomials: z(x,y)=(A 1 x 2 +A 2 x 4 +A 3 x 6 +â¦)+(B 1 y 2 +B 2 y 4 +B 3 y 6 +â¦)+(C 1 x 1 +C 2 x 3 +C 3 x 5 +â¦)+(D 1 y 1 +D 2 y 3 +D 3 y 5 +â¦). Linear terms, which represent a tilt of the basic surface, as well as terms with odd exponents, are usually omitted in practice. 10.1 Surfaces of Second-Order (Quadrics) The quadratic terms z(x,y)=A 1 x 2 +B 1 y 2 , taken for themselves, are contained in the surfaces of second order. They are also called quadrics in the 3D space (Quadric 3D) and have some importance for optics. These surfaces are aspherics themselves and therefore will be used as a basis for aspherics. Note that the sphere is also a surface of second order. They have the property that the intersection of any plane with them creates curves, which are called cone sections: ellipses, parabolas, hyperbolas, straight lines, and points. Straight lines and points are called degenerate cone sections in this context. The surfaces of second order are generally determined through the equation F(x,y,z)=a 11 +2a 12 x+2a 13 y+2a 14 z+2a 23 xy+2a 24 xz+2a 34 yz+a 22 x 2 +a 33 y 2 +a 44 z 2 =0, as result of the multiplication of a vector u T =(1,x,y,z) with a symmetrical 4 Ã 4 matrix A, that is, F(x,y,z)=u T Au=0, where A is the general matrix of the surfaces of second order.
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CHAPTER 10
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