CSG notation (InstanceTopic, 3)
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[ System created by [[Keiji]] ] | [ System created by [[Keiji]] ] | ||
'''Constructive Solid Geometry Notation''' is a way to represent any n-dimensional object. You start with a [[point (object)|point]], and perform each operation in order. | '''Constructive Solid Geometry Notation''' is a way to represent any n-dimensional object. You start with a [[point (object)|point]], and perform each operation in order. | ||
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| + | Important: CSG notation '''has been obsoleted''' by [[SSC notation]]. | ||
== Operations == | == Operations == | ||
These operations create the object. | These operations create the object. | ||
| - | * E = [[Extrude]]s the object. | + | *E = [[Extrude]]s the object. |
| - | * H = Does the same as E, but doesn't solidify the object. ''Obsolete, usage of C is preferred''. | + | *H = Does the same as E, but doesn't solidify the object. ''Obsolete, usage of C is preferred''. |
| - | * L = [[Lathe]]s the object. | + | *L = [[Lathe]]s the object. |
| - | * Q = Equivalent to M<sub>x</sub>[1]L, which effectively makes the object into a [[torus]]. | + | *Q = Equivalent to M<sub>x</sub>[1]L, which effectively makes the object into a [[torus]]. |
| - | * T = [[Taper]]s the object to a point such that the new lines are the same length as the original lines. | + | *T = [[Taper]]s 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. | + | *D = Tapers the object to two opposite points such that the new angles are 90 degrees. |
| - | * C<sub>''d''</sub> = | + | *C<sub>''d''</sub> = Gets the [[convex hull]] of the shape in ''d''-space. |
| - | * X<sub>''d''</sub>['' | + | *X<sub>''d''</sub>[''v'',''a''] = Truncates the ''d''-hypercells by v ∈ [0,1] default 0.5 at angle a ∈ (-1,1) default 0. |
| + | *P[a] = Projects all existing vertices onto a [[hypersphere]] of radius a ∈ (0,∞) default 1 in the dimension of the current [[bounding space]]. | ||
<br> | <br> | ||
These operations translate the object. A single operation for a matrix translation is being considered. | These operations translate the object. A single operation for a matrix translation is being considered. | ||
| - | * M<sub>''d''</sub>[''v''] = [[Translate]]s the object by ''v'' in dimension ''d''. | + | *M<sub>''d''</sub>[''v''] = [[Translate]]s the object by ''v'' in dimension ''d''. |
| - | * R<sub>''p''</sub>[''a''] = [[Rotate]]s the object by ''a'' in plane ''p''. | + | *R<sub>''p''</sub>[''a''] = [[Rotate]]s the object by ''a'' in plane ''p''. |
| - | * S<sub>''d''</sub>[''v''] = [[Stretch]]es the object by scale factor ''v'' in dimension ''d''. | + | *S<sub>''d''</sub>[''v''] = [[Stretch]]es the object by scale factor ''v'' in dimension ''d''. |
<br> | <br> | ||
These operations deal with groups of operations, and will separate groups if they are not explicitly defined with parentheses. | 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. | * * = Gets the [[cartesian product]] of the two groups. | ||
* # = [[Spherate]]s the two groups. | * # = [[Spherate]]s the two groups. | ||
| - | * & = [[Intersect]]s the two groups. | + | *& = [[Intersect]]s the two groups. |
| - | * + = [[Merge]]s the two groups. | + | *+ = [[Merge]]s the two groups. |
| - | * - = [[Subtract]]s the group on the right from the group on the left. | + | *- = [[Subtract]]s 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. | + | *( ) = Makes the contained groups into one group, so that subsequent operations apply to the whole contents. |
== Angle notation == | == Angle notation == | ||
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**Anything that isn't a literal will be treated as a number of radians, e.g. 2π or tan<sup>-1</sup>(2<sup>-1</sup>). | **Anything that isn't a literal will be treated as a number of radians, e.g. 2π or tan<sup>-1</sup>(2<sup>-1</sup>). | ||
*Zero degrees or radians means that the angle is half-way between one affected hypercell and another. | *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 | + | *They are ordered in the same order that dimensions are created using E, L, T and D operations. Normal order is xyzwφσ, 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 == | == Plane notation == | ||
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*EEEX<sub>1</sub>[<0>,(2+sqrt(2))<sup>-1</sup>] - gives the [[rhombicuboctahedron]]. | *EEEX<sub>1</sub>[<0>,(2+sqrt(2))<sup>-1</sup>] - gives the [[rhombicuboctahedron]]. | ||
*EEEX<sub>0</sub>[<0, 0>,(2+sqrt(2))<sup>-1</sup>] - gives the [[truncated cube]]. | *EEEX<sub>0</sub>[<0, 0>,(2+sqrt(2))<sup>-1</sup>] - gives the [[truncated cube]]. | ||
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Latest revision as of 21:11, 8 February 2014
[ System created by Keiji ]
Constructive Solid Geometry Notation is a way to represent any n-dimensional object. You start with a point, and perform each operation in order.
Important: CSG notation has been obsoleted by SSC notation.
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 Mx[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.
- Cd = Gets the convex hull of the shape in d-space.
- Xd[v,a] = Truncates the d-hypercells by v ∈ [0,1] default 0.5 at angle a ∈ (-1,1) default 0.
- P[a] = Projects all existing vertices onto a hypersphere of radius a ∈ (0,∞) default 1 in the dimension of the current bounding space.
These operations translate the object. A single operation for a matrix translation is being considered.
- Md[v] = Translates the object by v in dimension d.
- Rp[a] = Rotates the object by a in plane p.
- Sd[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 xyzwφσ, 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.
