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s2
go for the Reimann sphere
The Riemann sphere extends the complex plane by adding one point at Infinity. It is the plane plus one point, say it the North pole, with some good math formulas to ensure nice behaviuors, in particular to avoid functions discontinuity and other goodies like Conformal map i.e. circles are mapped into circles.
I dedicate this work to my father Umberto Casati and the great mathematician Luigi Bianchi who was a great inspiration for me.
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Index ¶
Constants ¶
This section is empty.
Variables ¶
This section is empty.
Functions ¶
Types ¶
type Circle ¶
type Circle struct {
A float64
C complex128
D float64
}
A Circle and a line are the same in this world. They are represented by an hermitian 2x2 matrix (a, b, c, d) hence b = cmplx.Conj(c) and a and d are real numbers. Lines have a = 0. The matrix determinant equals the negative of the radius square. When determinant is zero, the circle degenerates to a line. Imaginary circles exist, they have negative radius.
func NewCircle ¶
func NewCircle(center complex128, radius float64) Circle
NewCircle constructor. When radius is zero it will degenerate to a point, when radius is negative it will create an imaginary circle.
func NewLine ¶
func NewLine(p complex128, q complex128) Circle
NewLine constructor, returns a line that is the locus of points equidistant from the given p, q points.
type Point ¶
type Point struct {
Z, W complex128
}
A Point is represented by the couple p = (z, w) of two complex numbers where two points are equal if (z, w) = λ (z', w').
func LFT ¶
LFT computes a Linear fractional transformation (https://en.wikipedia.org/wiki/Linear_fractional_transformation).
func NewPoint ¶
func NewPoint(Z complex128) *Point
NewPoint is a Point constructor. It only takes one Z argument, since W defaults to 1, i.e. the point is already normalized.
Source Files