CSG notation (InstanceTopic, 3)

From Hi.gher. Space

[ System created by Rob ]

Constructive Solid Geometry Notation is a way to represent any N-Dimensional object. You start with a point, and perform each operation in order.

Operations

These operations create the object.

• E = Extrudes the object.
• H = Does the same as E, but doesn't solidify the object. Obsolete, usage of C is preferred.
• L = Lathes the object.
• Q = Equivalent to M_x[1]L, which effectively makes the object into a torus.
• T = Tapers the object to a point such that the new lines are the same length as the original lines.
• D = Tapers the object to two opposite points such that the new angles are 90 degrees.
• C_d = Connects all existing vertices such that the result is a non-curved non-intersecting d-frame polytope.
• X_d[a,v] = Truncates the d-hypercells by v at angle a. See below for angle notation.

These operations translate the object. A single operation for a matrix translation is being considered.

• M_d[v] = Translates the object by v in dimension d.
• R_p[a] = Rotates the object by a in plane p.
• S_d[v] = Stretches the object by scale factor v in dimension d.

These operations deal with groups of operations, and will separate groups if they are not explicitly defined with parentheses.

• * = Gets the cartesian product of the two groups.
• # = Spherates the two groups.
• & = Intersects the two groups.
• + = Merges the two groups.
• - = Subtracts the group on the right from the group on the left.
• ( ) = Makes the contained groups into one group, so that subsequent operations apply to the whole contents.

Angle notation

For use with the X operation.

• Angles are written within angle brackets < > and are separated with commas.
• Angles may be written in four different forms:
  • a b where a/b is the fraction of degrees out of 360, for example 90° = 1 4.
  • v where v is the pairing of a and b above.
  • x° where x is the number of degrees.
  • Anything that isn't a literal will be treated as a number of radians, e.g. 2π or tan^-1(2^-1).
• Zero degrees or radians means that the angle is half-way between one affected hypercell and another.
• They are ordered in the same order that dimensions are created using E, L, T and D operations. Normal order is xyzw, followed by unnamed dimensions, but to explicitly specify you can place a series of lowercase letters at the start of the string e.g. wxyzLLL.

Plane notation

For use with the R operation.

• Planes can be specified in two forms:
  • Provide two letters, eg. xw. This is shorter, but only gives cardinal planes.
  • Provide three points, eg. <0,0,0>,<1,0,0>,<0,0,1>. As long as the three points are not in the same line, this will give a plane that can be in any orientation.
• Note that in the R notation, the rotation happens around the hyperaxis that is perpendicular to the plane specified.

Examples

Examples of the C operation:

• EDDC₀ produces a nullframe octahedron. This is the same as xH+yH+zH.
• EDDC₁ produces a monoframe octahedron. This is the same as xyHD+xzHD+yzHD.
• EDDC₂ produces a diframe octahedron. This is the same as HDD.
• EDDC₃ produces a triframe octahedron. This is the same as EDD.

Examples of the X operation:

• EEEX₁[<0>,2^-1] - gives the octahedron. This is the same as EDD.
• EEEX₁[<0>,(2+sqrt(2))^-1] - gives the rhombicuboctahedron.
• EEEX₀[<0, 0>,(2+sqrt(2))^-1] - gives the truncated cube.

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