Coordinated Angle Spaces (CAS)

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Written By Adi Cox

9th July 2015

Geometry is typically about the properties of shapes in a space, but I am more interested in possible shapes of spaces themselves. A good mathematical way to look into the shapes of spaces is coordinate geometry.

I am looking at a simplified coordinate geometry that is only angles and radii of length one. The simplest example of this is the circumference of a unit circle. This is a one dimensional curved space that is a two dimensional subspace.

Notation:

In one dimension the circumference of a unit circle can be represented by x1(a1) where one dimensional space x1 is the line length that angle a tracks from angle zero along the circumference of the unit circle. So radians would be used for angles.

One dimensional 1 space example:

x1(pi/4) = the point on axis line x1 at length pi/4

Two dimensional 2 space example:

x1x2(pi/6) = the point in two dimensional space at:

x1 = sin(a1) = x2/hypotenuse = (1/2)/1 = point 1/2 on axis x1 x2 = cos(a1) = x1/hypotenuse = (square_root(3)/2)/1 = square_root(3)/2 = point square_root(3)/2 on axis x2

The general rule is that we can have the same number of axis as angles, written as;

x1(a1), x1,x2(a1,a2), x1,x2,x3(a1,a2,a3), x1,x2,x3,...xn(a1,a2,a3,...an)

We can have more angles than axis, written as;

x1(a1,a2), x1(a1,a2,a3), x1,x2(a1,a2,a3), x1,x2(a1,a2,a3,a4)

We can use any axis, written as

x2(a2), x2,x5(a2,a5), x1,x7(a1,a2,a4,a7)

The above diagram shows the x1,x2 points where axis x1 and x2 are perpendicular and meet at 0,0. The problem here is that where angle a1 and angle a2 are different by pi radians then we get coordinates 0,0.

If we use the axis x1 and x2 as the circumferences for unit circles with angles a1 and a2 respectively then there is only one 0,0 point where angles a1 and a2 are both zero. As shown in the diagram above.

The overlap occurs in standard x1,x2 cartesian coordinate system when the angles become greater than 2pi.

The 'Near Torus' coordinated angle space:

So called a Near Torus because instead of a hole for the torus we have point p.

The above picture is of a near Torus because point p is not a hole but a point where the surface of the near torus meet. This is because the inner circle on the x1,x2 plane is a unit circle, as are the cross section circles. That is unit circles that are across x1,x2,x3 cartesian space.

Angle a1 equals angle pi as it is the semincircle on the x1,x2 plane. Angle a2 is pi/2 because it is the angle from point p at the origin that fills the quarter of the cross section unit circle on the left. So what we have is a coordinate system of the surface of a near torus that uses just two angles a1,a2, where a1 and a2 are less than 2pi.

Converting 2 angles a1,a2 to 3 space cartesian x1,x2,x3:

For the near torus:

x1 = [1 + sin(a2)]sin(a1)x2 = [1 + cos(a2)]cos(a1)x3 = sin(a2)

where cos is -ve when pi/2 < a1 & a2 < pisin & cos are -ve when pi < a1 & a2 < (3pi)/2sin is -ve when (3pi)/2 < a1 & a2 < 2pi

For the torus:

x1 = [R + rsin(a2)]sin(a1)x2 = [R + rcos(a2)]cos(a1)x3 = rsin(a2)

where R is the large torus radius and r is the smaller torus radius.