Area-Preserving Piecewise Affine Mapping

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Area-Preserving Piecewise Affine Mapping

• Alan Saalfeld
• Ohio State University

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Area-preserving transformations

• A necessary and sufficient condition for any
  function (x,y) ? (u(x,y),v(x,y)) to preserve area
  is for
• the determinant of its Jacobian
• to be equal to (1). For an affine function, we
  have
• (x,y) ? (axbyc, dxeyf) (u(x,y),v(x,y))

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Affine transformations

• (x,y) ? (axbyc, dxeyf) (u(x,y),v(x,y))

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How Cartographers Show Distortion
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Affine transformations of the plane

• An affine function has the form
• (x,y) ? (axbyc, dxeyf)
• An affine function is a linear transformation
  followed by a translation
• The action on any 3 non-collinear points
  completely determines the affine map
• A triangle on three vertices maps to the triangle
  on the three image vertices.
• Lines go to lines
• Parallel lines go to parallel lines

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Describing Piecewise Affine Maps

• Piecewise affine interpolation is uniquely and
  implicitly defined BY SIMPLY DRAWING THE
  TRIANGULATION IN THE DOMAIN SPACE AND LABELING
  THE TRIANGLE VERTICES IN BOTH THE DOMAIN AND
  RANGE SPACES TO SHOW THE ASSOCIATION

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The composition of two Piecewise Affine maps is a
Piecewise Affine map
Invertible piecewise affine maps are also called
piecewise linear homeomorphisms (PLH maps).
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Any simple polygon can be mapped to any other
simple polygon by a PLH map with pre-specified
PLH boundary behavior

• Any n-sided polygon can be triangulated.
• Any triangulation of an n-sided polygon is also a
  triangulation of a similarly labeled regular
  n-gon.
• Every n-gon can be mapped by an invertible PLH
  map onto a regular n-gon.

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Can we control distortion more?

• Are there area-preserving PLH maps?
• Can we construct them to extend PLH maps on the
  boundary? YES! It simply means getting
  corresponding triangles to have the same area
• Can we find 1-continuous-parameter families of
  transformations that are area-preserving PLH maps
  for every parameter value? YES! These are
  zero-compression deformations.

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Can we simultaneously chop up two polygons into
triangles so that corresponding pairs of
triangles always have the same area ratio as the
ratio of the total areas of the original polygons?
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Decomposing quadrilaterals
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The same diagonal skew case
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Transformations of simple polygons

• PLH maps may be found that extend a boundary map
  and are area-proportional everywhere.
• Proof by induction
• Trivial for n3.
• Construction for n4.
• First show for convex sets for ngt4.

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Area-preserving transformations

• The following are equivalent
• 1. Any two convex n-sided polygons of equal area
  are homeomorphic under an area-preserving PLH map
  that is linear on each corresponding edge pair.
• 2. Any convex n-sided polygon is homeomorphic
  under an area-preserving PLH map to a regular
  n-gon of the same area. The homeomorphism may be
  taken to be linear on each corresponding edge
  pair.

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Node Splitting and Area Splitting

• Every convex polygon possesses a splitter that
  divides the area in equal parts and also splits
  the vertices into equal groups.

Proof This follows from the Ham Sandwich
Theorem applied to the area measure and the
counting measure on vertices. A more direct
proof for the particular two measures in question
is illustrated on the right.
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Area-preserving transformations

• Proof for convex sets
• Suppose 1. and 2. are true for all convex sets of
  size kltn, where ngt4.
• Find a splitter for a convex n-gon that splits it
  into two equal area (n/22)-sided convex
  polygons.
• Note n/2 2 lt n.
• Map each half into half of a regular n-gon.

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Area-preserving transformations

• A technical detail
• On 1 or 2 of the boundary edges, the ratio of the
  two pieces of the divided edge may differ for the
  convex (n/22)-gon and for the half of the
  regular n-gon. We can always adjust the edge
  pieces with yet another area-preserving PLH map
  so that they recover the original ratio.

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Area-preserving transformations

• Proof for all sets, convex or not
• Suppose any convex or non-convex k-gon for kltn
  has a PLH area-preserving, boundary-extending map
  to any same area convex k-gon, where ngt4.
• For the non-convex n-gon, triangulate and remove
  an ear. Ears always exist.
• Map the (n-1)-gon to a slab convex set with the
  edge from the missing ear going to the base edge
  of the slab. (A slab convex set has two
  consecutive acute angles at either end of the
  base edge.)
• Map the ear to an ear-sized triangle attached
  to the slab along the long edge. (Slab
  construction makes it possible.)

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Possible Extensions/Applications

• 3D
• Best quasi-conformal area-preserving piecewise
  affine transformation using no more than k
  Steiner points
• Zero-compression morphing

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Morphing

• Triangulation maps pi, qi, Tpi , pi ? qi
• Homotopies of triangulation maps pix0,1 ?R
• (pi,t) ? qi(t) (pi,0) ? qi(0)pi, (pi,1) ?
  qi(1)qi
• Cheap (faceted) morphing
• Barycentric coordinates pre-computed
• Homotopies in both domain and range
• pi(t) ? qi(t)
• Domain triangles and range triangles change in
  unison, maintaining relative size throughout
• At every stage, the transformation is
  area-preserving
• Area-preserving guarantees no singularities

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What is Zero-Compression Morphing?
It is continuous deformation that preserves area
(volume) everywhere at all times. If we can
identify corresponding pairs of simultaneously
changing quadrilaterals that have matching areas
at each stage throughout the deformation, then we
may add Steiner points and triangulate to produce
area-preservation everywhere.
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Zero-Compression Morphing
Example 1 A Rotating Raft
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Zero-Compression Morphing
Example 2 A Sliding Raft
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Zero-Compression Morphing
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Summary

• Easy-to-evaluate piecewise affine maps are fully
  described by a triangulation of the domain space
  and an assignment of an image to each vertex in
  the domain triangulation
• We can build a PLH map from one simple polygon
  onto any other simple polygon that has the same
  area-scale everywhere
• We may even build continuous 1-parameter families
  (homotopies) of PLH same-area-scale
  transformations.

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Zero-Compression Morphing
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Affine and piecewise affine transformations of
the plane

• There is uniform stretching (area-factor) within
  each piece of a piecewise affine transformation
• A PLH map defined on a triangle is
  area-preserving everywhere if and only if it
  sends the triangle to a triangle of the same
  area.

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Computing Barycentric Coordinates

• Every point in a triangle is a unique convex
  combination of the vertex points p1,p2,p3
• p?1p1?2p2?3p3
• where ?igt0, and 1?1?2?3.
• (?1,?2,?3) are called the barycentric coordinates
  of p with respect to p1,p2,p3.
• Converting to barycentric coordinates or back to
  Cartesian coordinates is easy.

?3 1-?1-?2
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Barycentric Coordinates,handy tool for affine
functions

• Every point in a triangle is a unique convex
  combination of the vertex points p1,p2,p3
• p?1p1?2p2?3p3
• where ?igt0, and 1?1?2?3.
• (?1,?2,?3) are called the barycentric coordinates
  of p with respect to p1,p2,p3.
• Converting to barycentric coordinates or back to
  Cartesian coordinates is easy.

?1

?2

?3 1-?1-?2
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Barycentric Coordinates are just Relative
Triangle Areas
?1

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Rubber-Sheeting Vector Interpolation with
Barycentric Coordinates

• Recall that every point inside a triangle is a
  unique convex combination of the vertex points
• p?1p1?2p2?3p3, where ?igt0, and1?1?2?3.

If v is a vector-valued function defined only at
p1, p2, and p3 , with v(pi)qi, then we may
define (extend) v(p) to be v(p)?1v(p1)?2v(p2)
?3v(p3) ?1q1?2q2?3q3, for the same ?1,
?2, and ?3. Sometimes these barycentric
coordinates are called weights.
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Piecewise affine transformations

• The interpolating function that preserves
  barycentric coordinates is precisely the affine
  transformation from the domain triangle onto the
  range triangle
• Barycentric coordinates can be computed for the
  domain (before the function itself is specified),
  then may be evaluated at run time (after the
  range values of the triangle vertices are
  specified)

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Isomorphic triangulations Piecewise Linear
Homeomorphisms

• Non-existence results
• Sometimes there just isnt any
• simultaneous triangulation on
• the given point sets
• Existence and complexity results
• Sometimes all triangulations work
• Sometimes some triangulations
• work, others do not
• If we allow ourselves to add some (O(n2)) point
  pairs, we can always find a simultaneous
  triangulation

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Triangulation maps Piecewise Linear Maps

• PL maps are described fully by their action on
  triangles
• PL maps are described fully by their action on
  vertices after domain triangles have been
  specified
• PL maps agree on shared edges that are straight
  line segments
• But we REALLY prefer our functions to be
  invertible (homeomorphisms)

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We can always find an isomorphic triangulation
(PLH map) of a set of point pairs if we allow
ourselves to add O(n2) additional point pairs.
p1
No
Yes