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GeoGebra

BOU 180 dimetric anamorphosis

Auteur :Frédéric ROBIN
This is a setting to represent a five-fold symmetric figure as a dimetric view of a 3D mesh. Required camera pitch angle is α = asin(tan(π/10)) ≈ 18.961° for this dimetric view instead of ≈ 35.264° for standard isometric view, so the additional rectangle on top labeled "Camera plane" has been rotated by the appropriate angles (yaw 45° + pitch α) and helps you rotate the view as required by clicking on it (or in classic mode using the "View in front of" command). In this setting the shapes that will be duplicated must be on the YZ plane (a slightly different setting could work with the XZ plane instead)
  • The original shape q1 (here the simple square from https://www.geogebra.org/m/zccvjdj8 has been replaced with a 8-points polygon that has a point symmetry; I wanted to keep both shapes in a single presentation with a button to switch between them but didn't find how to) is first duplicated four times with different orthogonal rotations to get the other (unchecked by default) yellow items q2 to q5
  • Then those five intermediates yellow shapes are reused to create the five final blue shapes q6 to q11 with different additional scales and rotations:
    • the two straightly vertical shapes q6 q7 are simply scaled along Z axis by factor s1
    • the two lateral inclined shapes q8 q9 are scaled along respectively X and Y axes by the same factor s1 and rotated about the same axes by angle β
    • the central inclined shape is first scaled along XY axes by factor s2 then has a steeper slope γ and its rotation and scale axes are diagonals so it is no more rectangular. Here I kept two versions of it:
      • Unchecked item q10 is a draft version of q11, using a more complicated formula that follows how I created it in my usual 3D software (Blender).
      • Checked item q11 is a smarter way to get the same result but it require a deformation matrix (shearing) that is not implemented in Blender geometry nodes.
      Note that all shapes appear as rectilinear polygons when seen from top.
Since the original shape q1 has a point symmetry, so does the final result so in this case we have an anamorphosis of the below tenfold symmetric star figure (without the kites). . Artistic use including kites and additional decorative polyhedra: https://skfb.ly/oVLBD .

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