Shape is an interesting property of objects because it is used in ordinary discourse in ways that seem to have little connection to how it is typically defined in mathematics. The present article describes how the concept of shape can be grounded within Euclidean and non-Euclidean geometry and also to human perception. It considers the formal methods that have been proposed for measuring the differences among shapes and how the performance of those methods compares with shape difference thresholds of human observers. It discusses how different types of shape change can be perceptually categorized. It also evaluates the specific data structures that have been used to represent shape in models of both human and machine vision, and it reviews the psychophysical evidence about the extent to which those models are consistent with human perception. Based on this review of the literature, we argue that shape is not one thing but rather a collection of many object attributes, some of which are more perceptually salient than others. Because the relative importance of these attributes can be context dependent, there is no obvious single definition of shape that is universally applicable in all situations.

- 1. The three-dimensional (3D) modeling software provides a set of primitive shapes, including a sphere, a cube, a cylinder, and a pyramid, which can all be deformed and/or combined to create an infinite variety of more complex shapes.
- 2. The old woman lived in a house, whose shape resembled a shoe.
- 3. All John has left from his boxing career is his misshapen nose.
- 4. We all recoiled at the grotesque shape of the creature, whose head was covered with small pointed horns and two writhing tentacles on each side.
- 5. The sculpture was shaped like a geographic surface with hills, dales, valleys, and ridges.

*Erlangen program*of Felix Klein

*group*of one-to-one transformations that map a space onto itself. The technical name for such a transformation is

*automorphism*(e.g., Anderson & Feil, 2015). The geometric structures that are defined for each geometry are those that are invariant under its associated automorphism group. He also noted that these geometries can be organized in a hierarchical manner so that the group associated with one geometry can be a subset of the group associated with another. Klein's proposal was more like a manifesto than a formal proof, and the hard mathematical work to develop the theory with full rigor was left to others, especially to his friend and collaborator Sophus Lie.

*isometry*—it preserves the lengths and angles on any geometric form. For example, to transform object A into object B in Figure 1, one must apply a translation and a rotation. It is easy to prove that two geometric figures are congruent (in the sense of having equal corresponding lengths and angles) if and only if there exists a transformation in the Euclidean group that makes the first figure identical to the second. If we are given an automorphism group, we can define congruence in terms of the equivalence classes that remain invariant with respect to arbitrary transformations in this group. In the case of the Euclidean group, all members of the same class have the same Euclidean shape and also the same absolute size. Additional invariants of Euclidean transformations include the perimeters and areas of 2D figures and the surface area and volumes of 3D objects.

*similarity*group. The latter contains the Euclidean group as a subgroup but also includes all uniform dilations. All three objects in Figure 1 belong in the same equivalence class with respect to the similarity relation. Like metric congruence, similarity preserves all angles. Whereas absolute distances, areas, and volumes are not preserved over similarity transformations, the ratios of these attributes remain invariant, as do any translational, rotational, or reflective symmetries. Many mathematicians maintain that the similarity relation captures well the literal meaning of the everyday English expression that two geometric figures “have the same shape.” Thus, all circles have the same shape, all spheres have the same shape, all cubes have the same shape, and so on.

*shape*per se, but they do provide a widely used definition of

*shape equivalence*. Two geometric figures

*have the same shape*if and only if one of them can be transformed into the other via a similarity transformation—that is, via some combination of translation, rotation, reflection, and uniform scaling. The notion of shape that is defined by these equivalence classes is referred to as

*Euclidean*(or, interchangeably,

*metric*) shape throughout this article to distinguish it from alternative notions of shape.

*affine geometry—*the study of the geometric properties that remain invariant under arbitrary

*affine transformations*. The latter also form a group—the

*affine group*of automorphisms. It contains the similarity group as a subgroup with the addition of nonuniform dilations, including shears. A shearing transformation is illustrated in the left panel of Figure 2. An appropriately chosen nonuniform dilation transforms a square into an arbitrary rectangle or rhombus. The equivalence relation defined by this group is more liberal than the similarity relation and partitions the space of geometric figures into coarser equivalence classes. For example, a circle is affine equivalent to an arbitrary ellipse. Thus, all ellipses (circles included) have the same

*affine shape*. Also, all triangles have the same affine shape. Angle measures in general (including perpendicularity) are not meaningful concepts in affine geometry, but parallelism is an affine invariant. Another important invariant of the affine automorphism group is the ratio of lengths of parallel line segments (see Figure 2), including bisection. (The relative lengths of nonparallel segments are not preserved.) Concavities, convexities, and planarity are also invariant under affine transformations. Straight lines remain straight and all incidence relations are preserved.

*projective geometry*. It is generated by the group of

*projective collineations*. A collineation is an automorphism that maps lines onto lines. If points A, B, and C are connected by a single line, then the projections of those points will also be connected by a single line. The set of all collineations forms a group that contains the affine group as a subgroup. In addition to collinearity and incidence, the so-called

*cross-ratio*of four points is also a projective invariant (see Figure 2).

**Objective measures of shape change**. There are many practical applications where it is important to quantify differences in shape, and there are many possible procedures by which this can be achieved. Because it is universally recognized that changes in the position, orientation, or size of an object have no effect on its objective shape, an ideal measure of shape change should not be affected by any of those transformations. Consider the two pairs of objects labeled A and B that are shown in Figure 6. The lower figure of pair A was created by displacing one vertex in a horizontal direction, and the lower figure of pair B was created by displacing an entire edge. In both cases, the original area of the figure is colored black and the changed region is colored red.

**Perceptual measures of shape change**. In a remarkable study published almost 40 years ago, Chen (1982) proposed a radical new hypothesis that the relative perceptual salience of different types of shape change may be systematically related to the Klein hierarchy of geometries. He referred to this as the topological approach to visual perception, based on his observations that changes to topological properties are easier to detect than any other type of shape change (see also Chen, 1982, 1983; He et al., 2015). Chen has provided numerous psychophysical examples to support this hypothesis, but he has never employed objective measures of shape change to see if any of them could potentially account for his results.

*On Growth and Form*. The method he employed for comparing 2D shapes involved covering them with a rectangular grid to provide a coarse coordinate system. He would then distort the grid on one so that the most salient landmarks on both shapes were located at corresponding positions on their respective grids (see Figure 8). The pattern of grid distortion required to make that happen highlighted the underlying geometric transformation by which the two forms were related.

*American Gothic*that is perceived to have a sad expression even though there is no downward curvature of the mouth. The right panel shows a transformed version of this face, in which the vertical distance between the brows and the mouth has been reduced. This causes the transformed face to appear angry.

*Theoretical Geography*, William Bunge (1962) proposed four criteria for evaluating any measure of shape: (1) It should be objective; (2) tt should not consist of something less than shape, such as a set of position coordinates; (3) it should not consist of something more than shape such as a set of parameters for a Fourier or Taylor series; and (4) it should not do violence to our intuitive notions of what constitutes shape. In order to satisfy this fourth criterion, any two objects that are similar according to a valid measure of shape should also be perceptually similar and vice versa.

*R*

^{2}values typically in excess of 0.97 (Egan & Todd, 2015; Todd, Norman, Koenderink, & Kappers, 1997). However, the results also show that the best-fitting surfaces are typically distorted relative to the ground truth by an affine stretching or shearing transformation in depth (e.g., see Figure 3), which is consistent with the known ambiguities of various sources of visual information such as shading or motion (Belhumeur, Kriegman, & Yuille, 1997; Koenderink & van Doorn, 1991; Todd & Bressan, 1990). These findings are also in agreement with the results obtained using global shape adjustment tasks (Bradshaw, Parton, & Glennerster, 2000; Johnston, 1991; Todd & Norman, 2003) or depth magnitude estimations (Loomis & Philbeck, 1999; Todd & Norman, 1991).

_{max}) is larger than in any other direction and another where the curvature (Κ

_{min}) is smaller than in any other direction. Koenderink noted that Κ

_{max}and Κ

_{min}can be transformed into two alternative measures: one called curvedness, which varies with scale, and another called the shape index, which is scale invariant (see Figure 20). The shape index partitions surface patches into five qualitatively distinct types: bumps, ridges, saddles, valleys, and dimples. These can define the nodes of a graph, and the adjacency relations between regions define the edges. These different types of curvature are easily identified in the iso-height contours of Figure 19. For bumps and dimples, the contours form closed loops, and for saddles, they diverge away from each other. One important advantage of this approach over the Morse theory analysis is that the shape index is invariant over variations in surface orientation, whereas local extrema of height, depth, or slant are not.

_{max}is a local maximum or K

_{min}is a local minimum. This is implicit in line drawings of polyhedra. The edges have infinite curvature, and the faces have no curvature at all. Of course this is impossible for real objects, but it is a reasonable approximation in many contexts. It is possible to generalize the concept of a sharp edge on polyhedral surfaces to include curvature extremal contours on more rounded surfaces, and the addition of those contours can dramatically enhance pictorial depictions. The lower left panel of Figure 21 shows the rim contours of the shaded image in the upper left panel together with its curvature extremal contours. Note how this produces a much more compelling depiction of the object's shape than what is obtained when the rim contours are presented in isolation.

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