Quadrays and Areaby Kirby Urner |
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If a and b have the same length d and the angle between them is 90 degrees, then our parallelogram is a square and its area is defined as d x d or "d squared". The game of quadrays includes xyz area among its library of methods but defaults to a different conceptualization of the process for deriving area from two vectors. |
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The area varies with the angle between two fixed-length vectors according to either method, as the resulting parallelogram or triangle changes shape. We will need a method to determine the angle between any two quadrays. One way to get this angle is to expand or reduce the lengths of both a and b to unity by applying a scalar of 1/distance. This will result in two vectors a' and b' with the same angle between them as between the original a and b, but both with length one: |
a' = a * 1/D(a) D(a') = 1 b' = b * 1/D(b) D(b') = 1 |
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The sin( ) function accepts an angle and outputs the corresponding ratio of oppositeSide/hypotenuse, while its inverse, arcsin( ), accepts this ratio as input and outputs the angle's measure. In other words: sin(angle/2) = D(a' - b')/2 angle = 2 * arcsin[D(a' - b')/2] |
One other consideration is that our native D( ) function returns
its length in radians, meaning in units equaling the radius of our
dense-packed ivm spheres e.g. |
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The function D(vector), developed in An Introduction to Quadrays, returns a vector's length or distance. Putting all of these facts together, we have enough information to design the required area methods: |
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The default area is equal to xyzArea multiplied by a constant, which is to be expected, since both vary linearly in proportion to edges a, b and the sine of the included angle. xyzArea = D(a)/2 * D(b)/2 * sin(angle) Area = D(a)/2 * D(b)/2 * sin(angle) * Root(4/3) However, we should remember another important difference: xyzArea refers to the area of a parallelogram, whereas the default method refers to the area of a triangle with half the area of the corresponding parallelogram. If we halve the xyzArea, then both numbers refer to the same geometric shape and we have the basis for a conversion constant. Area = K * xyzArea/2 K = 2 * Area/xyzArea K = 2 * Root(4/3)According to our default method, the concentric hierarchy icosahedron has an area of 20, the octahedron an area of 8, and the tetrahedron an area of 4. |
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