Abstract
3D geo-information analyses topological and metrical relationships between spatial objects. This analysis needs a suitable representation of the three-dimensional world. This paper proposes to use the 4D unit sphere as a model. In essence this model is already present in mathematical theories like Lie sphere geometry, Moebius geometry and Geometric Algebra. The forementioned theories use the stereographic projection implicitely to build the model. This paper explicitely uses this geometric transformation to introduce the model as simply as possible following both an intuitive geometric and a formal algebraic self-contained way. The calculation in a CAD-environment of 3D Voronoi cells around given 3D points gives a straightforward example of the topological and metrical capabilities of this model. The addition of geometrical meaningful algebraic operations to the model will increase its computational power.
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Notes
- 1.
Here rejection is used in the sense of ‘back’ projection. In Geometric Algebra the rejection of a vector has a different meaning: the component complementary to its projection on another vector.
- 2.
For a better symbolic discrimination of the elements in R n+1 in stead of the letter X the letter U is used for a general element, and the letter P denotes a pole. Consequently coordinates in R n+1 are denoted by \( {u_i} \).
- 3.
See also Sect. 4.5.
- 4.
See Sect. 2: substitute \( {u_i} = \displaystyle\frac{{{v_i}}}{{{v_{n + 2}}}} \) and scale to get a suitable representation.
- 5.
Historic aside: this is in fact the ancient construction of the mean proportional from Proposition 13 of book VI in The Elements of Euclid.
- 6.
Satz 6.54 from the Tractatus logico-philosophicus of Ludwig Wittgenstein.
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(Pim) Bil, W.L. (2011). Modeling Space by Stereographic Rejection. In: Kolbe, T., König, G., Nagel, C. (eds) Advances in 3D Geo-Information Sciences. Lecture Notes in Geoinformation and Cartography(). Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12670-3_2
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DOI: https://doi.org/10.1007/978-3-642-12670-3_2
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