XMAP: Five-Point Interpolating Map

doi:10.1016/j.cad.2021.103118

Computer-Aided Design

Volume 142, January 2022, 103118

https://doi.org/10.1016/j.cad.2021.103118 Get rights and content

Abstract

Xmap takes the unit square to a subset of a doubly curved surface. It interpolates five control-points, which are the images of the four corners and of the center of that square. Interactive tweaks of control-points produce unsurprising changes in the map. The Xmap is an extension of the BeCOTS map. When the control-points are coplanar, it is an exact generalization of the COTS map. We conjecture that Xmaps may prove useful for designing aesthetically pleasing 3D structures.

Section snippets

Prior art on 3-point and 4-point maps

Let $M (x, y)$ be a map that takes the unit-square domain $D = {[0, 1]}^{2}$ onto a range $R$ that is the subset of a surface $S$ . A d-point map is controlled by $d$ interpolation-constraints of the form $M (x_{i}, y_{i}) = p_{i}$ , for $i \in {1, 2 . . . d}$ . A map is planar if $S$ is a plane. Planar 3-point maps include the Affine [1] and the Moebius maps. Planar 4-point maps include the perspective [2], the FPI [3], and the COTS [4] maps. Non-planar 4-point maps include the bilinear and the BeCOTS [5] maps. For these,

Planar Xmap

The Xmap satisfies these five constraints: $M (0, 0) = a$ , $M (1, 0) = b$ , $M (1, 1) = c$ , $M (0, 1) = d$ , and $M (\frac{1}{2}, \frac{1}{2}) = e$ .

It may be defined as follows in terms of similarities. A SAS (Steadily Animated Similarity) [4], $S_{a, b, c, d}^{t}$ , takes point $p$ to $S_{a, b, c, d}^{t} \cdot p = f + m^{t} \vec{fp}^{\circ} (w t),$ with $m$ , $w$ , and $f$ computed as follows (from Eq. (8) of [4]). $S_{a, b, c, d}$ takes oriented edge $ab$ to oriented edge $dc$ . Hence, $m = | \vec{dc} | / | \vec{ab} |, w = {\vec{ab}}^{\lor} \vec{dc}, f = a + 〈 \vec{w} \cdot \vec{da}, \vec{w}^{⌞} \cdot \vec{da} 〉 / d,$ where $| \vec{u} |$ is the norm of $\vec{u}$ , $\vec{u}^{\lor} \vec{v}$ is the angle between $\vec{u}$ and $\vec{v}$ , $\vec{w}^{\circ} w$ is $\vec{w}$

Non-planar Xmap

In the non-planar Xmap, the control-points need not be coplanar (Fig. 4).

Since point $u$ (Section 2) lies in the plane through $a$ , $e$ , and $c$ . The parameters of $S_{a, e, e, c}$ may be computed¹ using: $m_{u} = | \vec{ec} | / | \vec{ae} |,$ $w_{u} = \vec{ae}^{\lor} \vec{ec},$ $\vec{n} = \underset{̲}{\vec{ec} \times \vec{ae}},$ $f = \vec{a} + m_{u} (cos (w_{u}) \vec{ae} + sin (w_{u}) \vec{ae} \times \vec{n}),$ where $\underset{̲}{\vec{v}}$ denotes the normalized version $\frac{1}{| \vec{v} |} \vec{v}$ of $\vec{v}$ .

$S_{b, e, e, d}$ may be computed similarly from $b$ , $e$ ,

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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There are more references available in the full text version of this article.

Cited by (1)

CTSP: CSG Combinations of Tran-Similar Two-Patterns of CSG Cells
2022, CAD Computer Aided Design
We propose a paradigm for the design and processing of solid structures that form a bent two-pattern of cells that are similar to each other, but have steadily varying scale and orientation. The proposed CTSP is a procedural representation that extends prior work on periodic lattices from a union of balls and beams, to a CSG combination of unions of cells, each of which is a CSG solid. To ensure consistent connections between neighboring cells, the primitives are defined implicitly in terms of two-patterns of balls. We propose an algorithm, which, in constant time, computes a set of ranges of cells that include all cells that interfere with a given query ball. We also propose an algorithm which enables interactive visualization of these structures, even when they have a large count of cells. We also discuss potential uses of this solution for modeling shelled lattices and multi-resolution periodic structures, where, for example, each cone-beam shell of a coarse shelled lattice is a CTSP.

^☆: This paper has been recommended for acceptance by Giuseppe Patane.

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XMAP: Five-Point Interpolating Map☆