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Universal Hyperbolic Geometry (UHG) module – quick reference
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Available commands:
BilinearForm Compute the inner product (bilinear form) of two vectors under the given metric.
BuildConicFromCanonical Build the conic expression (quadratic form) from a canonical matrix C.
Centroids Return the four centroid points (homogeneous) from the canonical matrix C.
CircumDualPoints Compute the dual points of the circumlines as a 3x4 matrix.
CircumJayPoints Compute the four CircumJay points (intersection of circumline and Jay line).
CircumWrenPoints Compute the four CircumWren points.
Circumcenters Compute the four circumcenters (points) from the canonical matrix C.
Circumcircles Return expressions for the four circumcircles (quadratic forms in x,y,z).
Circumlines Return the four circumlines (homogeneous lines) from the canonical matrix C.
ClassifyPoints Classify each point of the triangle as inside (-1), on (0), or outside (1) the null conic.
ClassifyTriangle Classify the triangle based on vertex positions relative to the null conic.
DetectGeometry Determine the type of geometry (Euclidean, Hyperbolic, Elliptic) from the metric matrix J.
Distance Compute the distance between two vertices (by index) of a triangle given its canonical matrix C.
DoubleTriangle Return the three vertices of the double triangle (polar triangle) from the canonical matrix C.
DualLine Compute the dual (polar) of a line with respect to the conic given by matrix C.
DualPoint Compute the dual (polar) of a point with respect to the conic given by matrix C.
FindOddPoint In a mixed triangle (some points inside, some outside), find the 'odd' point that differs from the other two.
GetInverseTransform Compute the inverse of the prepared triangle matrix (from PreparePoints).
GramMatrixUHG Compute the Gram matrix G = A * J * A^T for a matrix of points. This is the first step in normalising a triangle.
Info Display this help message. If no argument is given, list all commands with a short summary.
IntersectionPlot Plot the intersection curve of a surface F and a plane P in 3D.
JayLines Compute the four Jay lines from the canonical matrix C.
JayWrenPoints Compute the four JayWren points (intersection of Jay and Wren lines).
LineFromPoints Compute the homogeneous line through two points (cross product).
Medians Return the three median lines (homogeneous) from the canonical matrix C.
Midlines Compute the two midline lines (duals of the midpoints) with respect to the canonical conic.
Midpoints Compute the two midpoints (on the line) between vertices i and j of the triangle A.
NormalizePoints Normalise a triangle (3x3 matrix) to its canonical form. This is the main function for triangle analysis.
NormalizePrepared Normalise a triangle that has already been prepared with PreparePoints. Prints geometry type and triangle location.
OppositePoints Return the two points opposite to vertex i (the other two vertices).
Oppositelines Return the dual lines of the opposite points.
PlotAll2D Generate a 2D plot of the triangle and selected geometric elements in the Klein model (unit disk).
PlotAll3D Generate a 3D plot of the triangle and elements in projective or hyperboloid model.
PlotCircumlinear Project all objects onto the unit triangle plane x+y+z=1 (circumlinear coordinates).
PlotConic Plot a conic in the Klein model together with the null circle.
PlotConicOnUnitTriangle Plot the intersection of a conic with the unit triangle plane.
PlotHierarchy (Under development) Show the circumcenter hierarchy up to a given depth.
PlotSteps Print step‑by‑step instructions to manually build the circumlinear plot for a canonical matrix.
PlotStepsArray Generate a column of five plots showing step‑by‑step construction of the triangle analysis.
PlotUnitTriangle Plot the unit triangle (the simplex x+y+z=1) in 3D.
PlotUnitTriangleWithConic3D Comprehensive 3D plot of the unit triangle with a conic and optional UHG elements.
PointFromLines Compute the intersection point of two lines (cross product).
PointLocation Determine whether a point lies inside, on, or outside the null conic.
PreparePoints Scale and reorder the triangle so that it is ready for canonical normalisation.
ProjectiveQuadrance Compute the projective quadrance (an invariant) between two non‑null points.
Quadrance Compute the squared norm (quadrance) of a vector, which determines its position relative to the null conic.
ScalePoints Scale the triangle so that the quadrances of all points become ±1 (or 0 for ideal points).
ShowGeometry Print the detected geometry type and the metric matrix to the console.
SidePoints Compute the midpoints on the side opposite to vertex i.
Sideliness Compute the sidelines (duals of side points) opposite vertex i.
SoundConics Compute the four sound conics as coefficient lists (each list of 6 coefficients).
SoundPoints Compute the sound points from the tangent lines as a 3x4 matrix.
Spread Compute the spread (sin² of the angle) between two non‑null vectors.
Sydlines Compute the two sydlines (duals of the syndpoints) with respect to the canonical conic.
Syndpoints Compute the two syndpoints (orthogonal midpoints) between vertices i and j.
T_Axis Compute the T‑axis (a line) from the canonical matrix C.
TangentLines Compute the tangent lines at the circumdual points as a 3x4 matrix.
TransformConic Transform a conic matrix under the inverse prepared triangle (used in PlotConic).
TwinCircumcircles Return expressions for the four twin circumcircles (quadratic forms in x,y,z).
U_Axis Compute the U‑axis (a line) from the canonical matrix C.
V_Axis Compute the V‑axis (a line) from the canonical matrix C.
WrenLines Compute the four Wren lines from the canonical matrix C.
Use Info("command") for detailed help on a specific command.
Example: Info("NormalizePoints");
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J := Matrix([[1,0,0],[0,1,0],[0,0,-1]]):
a1 := [2/3, -3/2, 1]:
a2 := [-5/2, -3/2, 1]:
a3 := [1/2, 3, 5]:
A := Matrix([a1, a2, a3]):
res := NormalizePoints(A, J):
C := res["CanonicalMatrix"]:
dstcnc := BuildConicFromCanonical(C):
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Info("BuildConicFromCanonical");
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Purpose: Build the conic expression (quadratic form) from a canonical matrix C.
Parameters:
C : canonical matrix (from NormalizePoints)
Returns: polynomial expression in x,y,z
Example: dstcnc := BuildConicFromCanonical(C);
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Info("PlotConicOnUnitTriangle");
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Purpose: Plot the intersection of a conic with the unit triangle plane.
Parameters:
conicExpr : polynomial or equation
curveColor, curveThickness, showSurface, bereik
Returns: a 3D plot object
Example: PlotConicOnUnitTriangle(dstcnc, showSurface=true);
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PlotConicOnUnitTriangle(dstcnc);
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Info("PlotUnitTriangleWithConic3D");
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Purpose: Comprehensive 3D plot of the unit triangle with a conic and optional UHG elements.
Parameters:
conicExpr, planeExpr (default x+y+z-1),
colors, transparencies, sizes, thicknesses, ranges,
show_legend, plot_width, plot_height, legend_width, legend_height,
elements (same as PlotAll2D plus "conic")
Returns: a 3D plot object (with optional legend).
Example: PlotUnitTriangleWithConic3D(dstcnc, x+y+z-1, elements=["conic","vertices"]);
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PlotUnitTriangleWithConic3D(dstcnc, x+y+z-1, elements=["triangle","conic","vertices"],vertexColor ="red");
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J := Matrix([[1,0,0],[0,1,0],[0,0,-1]]):
a1 := [2/3, -3/2, 1]:
a2 := [-5/2, -3/2, 1]:
a3 := [1/2, 3, 5]:
A := Matrix([a1, a2, a3]):
res := NormalizePoints(A, J):
C := res["CanonicalMatrix"]:
dstcnc := BuildConicFromCanonical(C):
PlotUnitTriangleWithConic3D(dstcnc, x+y+z-1,
elements = ["all"],
(*showSurface = true,*)
show_legend = true);
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PlotUnitTriangleWithConic3D(dstcnc, x+y+z-1,
triangleColor = "yellow",
curveColor = "cyan",
elements = ["conic","vertices"],
showSurface = true,
show_legend = true);
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PlotUnitTriangleWithConic3D(dstcnc, x+y+z-1,
triangleColor = "yellow",
curveColor = "cyan",
elements = ["triangle","conic","vertices"],
(*showSurface = true,*)
show_legend = true);
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