Category Archives: Lecture

Conformal maps III: Stereographic Projection

Stereographic projection σ:RnSn{n} where n=(0,,0,1)Rn+1 is the northpole of Sn is a special case of an inversion: Let us consider the hypersphere SRn+1 with center n and radius r=2 and look at the image of \[\mathbb{R}^n =\left\{\mathbf{x}\in\mathbb{R}^{n+1}\,{\large … Continue reading

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Conformal Maps II: Inversions

Let MRn be a domain. A smooth map f:MRn is called conformal if there is a smooth function ϕ:MR and a smooth map A from M into the group O(n) of orthogonal n×n-matrices such that … Continue reading

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Quaternions

Quaternions, H are a number system like real or complex numbers but with 4 dimensions. In particular, H is nothing but R4 together with a multiplication law. The identification of H and R4 is given by: $$ \mathbb{H} = \lbrace … Continue reading

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Conformal Maps I: Holomorphic Functions

If MR2 is a plane domain and the image of parametrized surface f:MR3 is contained in R2={(x,y,z)R3|z=0} then the defining equations of a conformal map |fu|=|fv|fu,fv=0 imply that |fv| arises from |fv| by … Continue reading

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Conformal Parametrizations of Surfaces

In the context of surfaces the strict analog of arclength parametrized curves is an isometric immersion f:MR3 of a standard surface into R3. Here “isometric” means that lengths of curves and intersection angles of curves on the surface … Continue reading

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Arclength-Parametrized Curves

A parametrized curve γ[0,L]R3 is called parametrized by arclength provided that γ(t) moves with unit speed if we interpret t as time: |γ(t)|=1  for all  t[0,L]. Sampling an arclength-parametrization γ at evenly spaced points t=mLn for integer values of mContinue reading

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Sampled Parametrized Curves

In the last post we created geometric objects from scratch, including the underlying combinatorics. In most situations it is much more convenient to start with already existing geometry and transform it. This approach is similar to the standard way of … Continue reading

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Creating Geometry From Scratch

Here we explain how to generate geometry procedurally. There are two node types that allow tho do this: Nodes of type Attribute Wrangle allow to specify geometry using the programming language VEX. Nodes of type Python allow to do the same in the … Continue reading

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Combinatorial Geometry in Houdini

As primitive geometric objects (objects that do not arise as combinations of other objects) Houdini supports also round spheres, cylinders (called tubes in Houdini), volumes and other things. We will focus here on Houdini’s implementation of the combinatorial complexes described … Continue reading

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Combinatorial Geometry: Simplicial Complexes

While (for good reasons) we have restricted our treatment of combinatorial cell complexes to the two-dimensional case, the theory of n-dimensional simplicial complexes is rather straightforward: Definition: A simplicial complex is a finite set P together with a set SContinue reading

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