# Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation

^{*}

## Abstract

**:**

## 1. Introduction

_{i}and q

_{i}in the 2D image. There is always a pair of 3D points P

_{i}and Q

_{i}, whose images are p

_{i}and q

_{i}, such that P

_{i}and Q

_{i}are symmetric with respect to some plane. The symmetry plane bisects the line segment P

_{i}Q

_{i}and is orthogonal to this segment. There are infinitely many such solutions. The individual theorems specify the family of these solutions for perspective and orthographic projections and show that if the image curves are continuous, the 3D symmetric interpretations are also continuous, for both one-to-one and many-to-many point correspondences. First, we provide a proof for a simple case where there is a unique correspondence of pairs of symmetric points. We then generalize the theorems to the case of multiple correspondences. In Section 3, we discuss the role of symmetry in contour-configural organization, as well as the role of other constraints in detecting symmetry from a single 2D image.

## 2. Theorems and Proofs

_{s}, whose normal is n

_{s}(n

_{x}, n

_{y}, n

_{z}). Let P

_{i}(x

_{Φi}, y

_{Φi}, z

_{Φi}), be a point on Φ and Q

_{i}(x

_{Ψi}, y

_{Ψi}, z

_{Ψi}) be its symmetric counterpart on Ψ. Symmetry line segments, which are line segments connecting pairs of corresponding points on Φ and Ψ are parallel to the normal of Π

_{s}in the 3D space. Perspective images of these lines intersect at the vanishing point v on the image plane. The 3D orientation of Π

_{s}is specified by its slant σ

_{s}and tilt τ

_{s}. Without restricting generality, assume τ

_{s}= 0. Note that when τ

_{s}is not zero, we can always rotate the 3D coordinate system around the z-axis by τ

_{s}. Under this assumption, the normal to the symmetry plane is n

_{s}(n

_{x}, 0, n

_{z}). The slant of the symmetry plane is σ

_{s}= atan(n

_{x}/n

_{z}).

_{I}and the x- and y-axes of the 3D Cartesian coordinate system be the 2D coordinate system on the image plane. Let the center of projection F be on the positive side of the z-axis of the 3D Cartesian coordinate system: z

_{f}> 0 where z

_{f}is a z-value of F.

_{s}are finitely long and tame:

**Definition**

**1:**

^{2}arcs that have following properties; each arc is twice continuously differentiable and a tangent line at every non-endpoint of the arc does not have any intersection with the arc.

#### 2.1. A Pair of 2D Curves with Unique Correspondences—The Case of a Perspective Projection

**Theorem**

**1:**

_{φ}

_{0}and e

_{φ}

_{1}, and the endpoints of ψ be e

_{ψ}

_{0}and e

_{ψ}

_{1}. Assume that the lines e

_{φ}

_{0}e

_{ψ}

_{0}and e

_{φ}

_{1}e

_{ψ}

_{1}intersect at a point v that (i) is not on φ or ψ and (ii) is not between e

_{φ}

_{0}and e

_{ψ}

_{0}or between e

_{φ}

_{1}and e

_{ψ}

_{1}. Additionally, assume that each half line that emanates from v and intersects φ has a unique intersection with ψ and vice versa (see Figure 3). Then, for a given center of projection F there exists a pair of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is a perspective projection of Φ and ψ is a perspective projection of Ψ.

**Proof:**

**Figure 3.**F = [0, 0, z

_{f}] is the center of perspective projection and Π

_{I}(z = 0) is the image plane. φ and ψ are two given curves on the image plane. e

_{φ}

_{0}and e

_{φ}

_{1}are the endpoints of φ, and e

_{ψ}

_{0}and e

_{ψ}

_{1}are the end points of ψ. The lines e

_{φ}

_{0}e

_{ψ}

_{0}and e

_{φ}

_{1}e

_{ψ}

_{1}intersect at point v on the x-axis. A line that is emanating from v and intersects with φ has a unique intersection with ψ and vice versa.

_{s}. That is, the line segment connecting the backprojected points is bisected by Π

_{s}and parallel to the normal of Π

_{s}. It will be also shown that the backprojected points form a pair of continuous curves.

_{v}, 0, 0]. We express φ and ψ in a polar coordinate system (r, α), where r is the distance from the vanishing point v and α is the angle measured relative to the direction of the x-axis. Then, the point p

_{i}= [x

_{φi}, y

_{φi}, 0] = [x

_{v}+ r

_{φ}(α

_{i})cosα

_{i}, r

_{φ}(α

_{i})sinα

_{i}, 0] on φ and the point q

_{i}= [x

_{ψi}, y

_{ψi}, 0] = [x

_{v}+ r

_{ψ}(α

_{i})cosα

_{i}, r

_{ψ}(α

_{i})sinα

_{i}, 0] on ψ are corresponding. Note that both r

_{φ}(α) and r

_{ψ}(α) are continuous functions and they are always positive (r

_{φ}(α), r

_{ψ}(α) > 0). Let the equation of the symmetry plane Π

_{s}be: P

_{i}and Q

_{i}, the 3D inverse perspective projections of p

_{i}and q

_{i}, are symmetric with respect to Π

_{s}if and only if they satisfy the following two requirements: the line segment connecting P

_{i}and Q

_{i}is parallel to the normal of Π

_{s}and is bisected by Π

_{s}.

_{i}and Q

_{i}is parallel to the normal of the plane Π

_{s}: Note that in an inverse perspective projection, an image point [x, y, 0] projects to a 3D point [x(z

_{f}– z)/z

_{f}, y(z

_{f}– z)/z

_{f}, z]. Hence, P

_{i}= [(z

_{f}– z

_{Φi})(x

_{v}+ r

_{φ}(α

_{i})cosα

_{i})/z

_{f}, (z

_{f}– z

_{Φi})r

_{φ}(α

_{i})sinα

_{i}/z

_{f}, z

_{Φi}] and Q

_{i}= [(z

_{f}– z

_{Ψi})(x

_{v}+ r

_{ψ}(α

_{i})cosα

_{i})/z

_{f}, (z

_{f}– z

_{Ψi})r

_{ψ}(α

_{i})sinα

_{i}/z

_{f}, z

_{Ψi}]. Then, combining (1) and (2), we obtain: From Equation (3), we obtain the following three facts. First, –d/c is an intersection of the symmetry plane Π

_{s}and the z-axis; it specifies the position of Π

_{s}. Second, the normal to this plane is [– x

_{v}/z

_{f}, 0, 1], which is parallel to a vector [x

_{v}, 0, – z

_{f}] connecting the center of perspective projection with the vanishing point. This immediately follows from the fact that v is the vanishing point corresponding to the lines connecting the pairs of 3D symmetric points, which are all normal to the symmetry plane. Third, a vanishing line (horizon) h of Π

_{s}is parallel to the y-axis on the image plane Π

_{I}. The line h intersects x-axis at: The next equation represents the fact that the line segments connecting pairs of 3D symmetric points are bisected by the symmetry plane. Let M

_{i}be the midpoint between P

_{i}and Q

_{i}–the midpoint lies on the symmetry plane Π

_{s}: From Equations (2) and (5), a perspective projection of M

_{i}to the image plane Π

_{I}can be written as follows: Equation (6) shows that m

_{i}is on a 2D line segment p

_{i}q

_{i}and is determined only by 2D image features on Π

_{I}. It does not depend on the position of the center of projection F. Recall that both r

_{φ}(α) and r

_{ψ}(α) are continuous functions and they are always positive (r

_{φ}(α), r

_{ψ}(α) > 0). It follows that the midpoints of the corresponding pairs of points on φ and ψ form a 2D continuous curve between φ and ψ on Π

_{I}. From Equations 2–6, we have: It is obvious that (7) and (8) represent continuous functions unless m

_{i}is on h: x

_{mi}= x

_{h}. Using Equations (6) and (4), we can rewrite x

_{mi}= x

_{h}as follows: Note that the left-hand side of Equation (9) is a cross-ratio [x

_{v}+ r

_{φ}(α

_{i})cosα

_{i}, x

_{v}+ r

_{ψ}(α

_{i})cosα

_{i}; x

_{v}, x

_{h}] = [x

_{φi}, x

_{ψi}; x

_{v}, x

_{h}]. If x

_{mi}= x

_{h}, z

_{Φi}and z

_{Ψi}diverge to ±∞ and Φ and Ψ are not continuous. This is because a projecting line emanating from F and going through m

_{i}does not intersect Π

_{s}. As a result, M

_{i}, which should be a midpoint between P

_{i}and Q

_{i}, cannot be determined. Recall that 2D projections of midpoints of 3D symmetric pairs of points form a 2D continuous curve on Π

_{I}. Hence, this curve must not have any intersection or tangent point with h. The whole curve must be either to the left or right of h. It follows that the denominators in (7) and (8) must be always positive or always negative for given φ and ψ. If this criterion is not satisfied, the 3D curves will not be continuous. Note that if the position of the center of projection F is a free parameter (this happens when the camera is uncalibrated), it is always possible to set F and thus h so that the criterion for continuity will be satisfied because the curve connecting the midpoints does not depend on F.

_{f}– z)/z

_{f}, y(z

_{f}– z)/z

_{f}, z]. Assume that d/c ≠ – z

_{f}—otherwise, the 3D interpretation will be degenerate with all the 3D points coinciding with the center of perspective projection F (Except for the case when the symmetry plane coincides with the YOZ plane. This can happen when the image curves are themselves symmetric). It can be seen that d/c determines the size, but not the shape of the 3D curves Φ and Ψ; z

_{f}+ d/c is a scale factor with respect to F as a center of scaling. Recall that the denominators in (7) and (8) must be always positive or always negative. From Equations (7) and (8), d/c can be adjusted so that Φ and Ψ are in front of the center of projection F and the image plane Π

_{I}. The symmetric pair of 3D curves produced from the curves in Figure 3 using Equations (7) and (8) is shown in Figure 4.

_{s}.

#### 2.2. A Pair of 2D Curves with Unique Symmetric Correspondences—the Case of an Orthographic Projection

_{I}intersect at a vanishing point v. Since v is an intersection of Π

_{I}and a line which emanates from F and is parallel to n

_{s}, the position of v is [x

_{v}, 0, 0] = [–z

_{f}tanσ

_{s}, 0, 0]. The sine and cosine of the slant σ

_{s}of the symmetry plane Π

_{s}can be expressed as follows: where L is the distance between the vanishing point v and the center of projection F: Let P

_{i}= [x

_{Φi}, y

_{Φi}, z

_{Φi}], be a point on Φ and Q

_{i}= [x

_{Ψi}, y

_{Ψi}, z

_{Ψi}] be its symmetric counterpart on Ψ. Recall that the symmetry line segments are parallel to the normal n

_{s}of the symmetry plane Π

_{s}and n

_{s}= [sinσ

_{s}, 0, cosσ

_{s}]. Hence, y

_{Φi}= y

_{Ψi}. Let, p

_{i}= [x

_{φi}, y

_{φi}, 0] be a perspective image of P

_{i}and q

_{i}= [x

_{ψi}, y

_{ψi}, 0] be a perspective image of Q

_{i}in Π

_{I}. A line segment connecting P

_{i}and Q

_{i}is a symmetry line segment and a line segment connecting p

_{i}and q

_{i}is a projected symmetry line segment; recall that a projected symmetry line segment intersects the x-axis at v. The p

_{i}and q

_{i}were represented in a polar coordinate system and written as p

_{i}= [x

_{φi}, y

_{φi}, 0] = [x

_{v}+ r

_{φi}cosα

_{i}, r

_{φi}sinα

_{i}, 0] and q

_{i}= [x

_{ψi}, y

_{ψi}, 0] = [x

_{v}+ r

_{ψi}cosα

_{i}, r

_{ψi}sinα

_{i}, 0], where r

_{φi}and r

_{ψi}are the distances of p

_{i}and q

_{i}from v when α = α

_{i}. Note that the 3D points P

_{i}and Q

_{i}and their 2D projections p

_{i}and q

_{i}satisfy Equations (7) and (8). From Equations (7), (10) and (11), we obtain z

_{Φi}(an analogous formula can be written for z

_{Ψi}): Recall that an orthographic projection is produced from a perspective projection by moving the center of projection F to infinity: z

_{f}→ +∞. As z

_{f}goes to infinity, L goes to positive infinity, as well. From Equation (12), the limit of z

_{Φi}as z

_{f}goes to infinity is: The limit of z

_{Ψi}is obtained in an analogous way: Recall that in an inverse perspective projection, an image point [x, y,0] projects to a 3D point [x(z

_{f}– z)/z

_{f}, y(z

_{f}– z)/z

_{f}, z]. As z

_{f}goes to infinity, the limit of [x(z

_{f}– z)/z

_{f}, y(z

_{f}– z)/z

_{f}, z] is [x, y, z], which is an inverse orthographic projection of [x, y, 0].

_{v}goes to negative infinity as z

_{f}goes to infinity; it follows that all α

_{i}become zero. This means that the projected symmetry line segments become parallel to one another and to the x-axis, and the vanishing point v goes to infinity. Hence, the slant σ

_{s}of the symmetry plane cannot be computed from the 2D image; instead, σ

_{s}becomes a free parameter in the 3D interpretation under an orthographic projection. It follows that there are infinitely many 3D symmetric curves that are consistent with a pair of 2D curves φ and ψ. In other words, the 3D curves form a one-parameter family characterized by σ

_{s}; σ

_{s}changes the aspect ratio and the orientation of the 3D shapes of the curves Φ and Ψ [10]. Note that if sin2σ

_{s}= 0 (σ

_{s}is 0 or 90°), z

_{Φi}and z

_{Ψi}diverge to ±∞. Hence, σ

_{s}should not be 0 or 90°. These two cases correspond to degenerate views of Φ and Ψ. When σ

_{s}is 0°, the symmetry plane is parallel to the image plane; φ and ψ will then coincide with each other in the 2D image. In such a case, the 3D recovery of a pair of symmetric 3D curves becomes trivial: one produces any Φ from ϕ and then Ψ is obtained as a mirror reflection of Φ. When σ

_{s}is 90°, the symmetry plane is perpendicular to the image plane. In such a case, φ and ψ, themselves, must be mirror symmetric in the 2D image in order for the 3D symmetric interpretation to exist. But then, the 2D curves themselves represent one possible 3D symmetric interpretation. The ratio d/c is another free parameter, but it only changes the position of Φ and Ψ along the z-axis and does not change their 3D shapes or orientations. From these results, Theorem 1 for a perspective projection generalizes to Theorem 2 for an orthographic projection.

**Theorem**

**2:**

_{φ}

_{0}and e

_{φ}

_{1}, and the endpoints of ψ be e

_{ψ}

_{0}and e

_{ψ}

_{1}. Assume that φ and ψ have the following properties: (i) e

_{φ}

_{0}e

_{ψ}

_{0}||e

_{φ}

_{1}e

_{ψ}

_{1}and (ii) a line that is parallel to e

_{φ}

_{0}e

_{ψ}

_{0}and intersects with φ has a unique intersection with ψ and vice versa (see Figure 5). Then, there exist infinitely many pairs of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space, such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is an orthographic projection of Φ and ψ is an orthographic projection of Ψ.

**Proof:**

**Figure 5.**φ and ψ are two 2D curves. e

_{φ}

_{0}and e

_{φ}

_{1}are the endpoints of φ, and e

_{ψ}

_{0}and e

_{ψ}

_{1}are the end points of ψ. The lines e

_{φ}

_{0}e

_{ψ}

_{0}and e

_{φ}

_{1}e

_{ψ}

_{1}are parallel to the x-axis and do not have any intersection with φ and ψ. A line that is parallel to e

_{φ}

_{0}e

_{ψ}

_{0}and intersects with φ has a unique intersection with ψ and vice versa.

_{φ}

_{0}e

_{ψ}

_{0}and e

_{φ}

_{1}e

_{ψ}

_{1}be horizontal. This does not restrict the generality: if these line segments are not horizontal, we rotate the image so that they become horizontal. For any point p

_{i}= [x

_{φi}, y

_{φi}] on φ, we find its counterpart on ψ as q

_{i}= [x

_{ψi}, y

_{ψi}]. Note that q

_{i}is found as an intersection of ψ and a horizontal line y = y

_{φi}. Hence, y

_{φi}= y

_{ψi}. We assume that this intersection is unique. Then, both φ and ψ can be represented as functions of y: x

_{φi}= x

_{φ}(y

_{φi}) and x

_{ψi}= x

_{ψ}(y

_{φi}). From Equations (13) and (14), the 3D symmetric curves Φ and Ψ are produced by computing the positions of all points P

_{i}and Q

_{i}as follows: where σ

_{s}is a slant of the symmetry plane. The tilt τ

_{s}of the symmetry plane is zero. Equations (15) and (16) allow one to compute a pair of 3D symmetric curves Φ and Ψ from a pair of 2D curves φ and ψ under an orthographic projection. It is obvious from these equations that Φ and Ψ are continuous when φ and ψ are continuous. Recall that slant σ

_{s}is a free parameter; it can be arbitrary, except for sin2σ

_{s}= 0. So, the 3D symmetric curves form a one-parameter family characterized by σ

_{s}. Equations (15–16) show that a relation between the pair of 2D curves φ and ψ and the pair of 3D curves Φ and Ψ becomes computationally simple when σ

_{s}is 45° (or – 45°) and d/c is 0; the absolute value of the z-coordinate of Q

_{i}is equal to the x-coordinate of P

_{i}and the absolute value of the z-coordinate of P

_{i}is equal to the x-coordinate of Q

_{i}. The symmetric pair of 3D curves produced using Equations (15) and (16) and consistent with φ and ψ in Figure 5 is shown in Figure 6.

#### 2.3. A Pair of 2D Curves with Multiple Symmetric Correspondences

**Figure 7.**An asymmetric pair of 2D curves with multiple symmetric correspondences and its 3D symmetric interpretation. (

**a**) An asymmetric pair of 2D curves. Some points of the right curve correspond to three points of the left curve. These curves can be still interpreted as a 2D orthographic projection of a symmetric pair of 3D curves. The slant σ

_{s}of the symmetry plane was set to 35°; (

**b**) Three different views of the 3D symmetric interpretation produced from the pair of the 2D curves in (

**a**). The numbers in the bottom are the values of the slant σ

_{s}of the symmetry plane of the symmetric pair of the 3D curves. For σ

_{s}equal to 35°, the image is identical to that in (

**a**). When σ

_{s}is 90°, its 2D projection itself becomes symmetric. See Demo 5 in supplemental material for an interactive illustration of the 3D symmetric curves [2].

**Theorem**

**3:**

_{φ}

_{0}and e

_{φ}

_{1}, the endpoints of ψ be e

_{ψ}

_{0}and e

_{ψ}

_{1}. Let a line connecting e

_{φ}

_{0}and e

_{ψ}

_{0}be l

_{0}and that connecting e

_{φ}

_{1}and e

_{ψ}

_{1}be l

_{1}. Assume that φ and ψ have the following properties: (i) l

_{0}||l

_{1}, (ii) l

_{0}and l

_{1}do not have any intersection with φ and ψ and (iii) a line that is parallel to l

_{0}and intersects φ has one or a finite number of intersections with ψ and vice versa (see Figure 8). Then, there exist infinitely many pairs of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space, such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is an orthographic projection of Φ and ψ is an orthographic projection of Ψ.

**Proof:**

**Figure 8.**φ and ψ are 2D curves. e

_{φ}

_{0}and e

_{φ}

_{1}are the endpoints of φ, and e

_{ψ}

_{0}and e

_{ψ}

_{1}are the end points of ψ. l

_{0}is a line connecting e

_{φ}

_{0}and e

_{ψ}

_{0}, and l

_{1}is a line connecting e

_{φ}

_{1}and e

_{ψ}

_{1}. l

_{0}and l

_{1}are parallel to the x-axis and do not have any intersection with φ and ψ. A line that is parallel to l

_{0}and intersects with φ has one or more intersections with ψ and vice versa.

_{s}and produce a symmetric pair of continuous curves.

_{φ}

_{0}e

_{ψ}

_{0}and e

_{φ}

_{1}e

_{ψ}

_{1}are horizontal. In this case, e

_{φ}

_{0}, e

_{φ}

_{1}, e

_{ψ}

_{0}and e

_{ψ}

_{1}are either global maxima or minima of φ and ψ along a vertical axis on the 2D image. Consider horizontal lines that are tangent to the curves at their local extrema (Figure 8). Intersections and tangent points of these horizontal lines with φ and ψ are labeled by numbers sequentially along each curve; p

_{1}, …, p

_{m}on φ and q

_{1}, …, q

_{n}on ψ (in Figure 8, n = 15, and m = 13). Both φ and ψ are divided into segments p

_{1}p

_{2}, p

_{2}p

_{3}etc. Let the endpoints e

_{φ}

_{0}, e

_{φ}

_{1}, e

_{ψ}

_{0}and e

_{ψ}

_{1}on φ and ψ be p

_{0}, p

_{m}

_{+1}, q

_{0}and q

_{n}

_{+1}, respectively. Let c

_{φi}be a segment of ϕ connecting p

_{i}and its successor p

_{i}

_{+1}. Then, c

_{φi}is a curve that is monotonic and continuous between p

_{i}and p

_{i}

_{+1}and these two points are the endpoints of c

_{φi}. Similarly, let c

_{ψj}be a segment of ψ connecting q

_{j}and its successor q

_{j}

_{+1}. A segment c

_{φi}of φ has, at least, one corresponding segment of ψ; the endpoints of these segments of ϕ and ψ form parallel line segments. From Theorem 2, a pair of c

_{φi}and each of the corresponding segments of ψ are consistent with an infinite number of 3D symmetric interpretations under an orthographic projection; the one-parameter family of symmetric pairs of 3D curves is characterized by the slant of a symmetry plane. The tilt of the symmetry plane is zero and its depth along the z-axis is arbitrary. It follows that, among all corresponding pairs of 2D segments of these curves, their possible 3D symmetric interpretations can share a common symmetry plane Π

_{s}with some slant σ

_{s}and depth. Hence, φ and ψ are consistent with a one parameter family of symmetric pairs of the 3D fragmented curves that are backprojections of the 2D segments of the 2D curves. In order to prove Theorem 3, we show that, for each member of the family, the symmetric pairs of the 3D fragmented curves produce a symmetric pair of 3D continuous curves whose endpoints are backprojections of the endpoints of φ and ψ. An orthographic projection of such a symmetric pair of the 3D continuous curves will be φ and ψ.

_{0}, …, p

_{m}

_{+1}on φ. The columns represent the points q

_{0}, …, q

_{n+}

_{1}on ψ. A circle at (p

_{i}, q

_{j}) represents a corresponding pair of points p

_{i}and q

_{j}in Figure 8. These circles are nodes in a graph representing the possible correspondences among pairs of points in Figure 8. Let an edge in this graph connecting (p

_{i}, q

_{j}) and (p

_{k}, q

_{l}) be labeled (p

_{i}, q

_{j})-(p

_{k}, q

_{l}). Note that the edge (p

_{i}, q

_{j})-(p

_{k}, q

_{l}) is the same as (p

_{k}, q

_{l})-(p

_{i}, q

_{j}). The edge (p

_{k}, q

_{l})-(p

_{i}, q

_{j}) represents a pair of segments of ϕ and ψ, such that c

_{φ}

_{min(i,k)}connects p

_{i}and p

_{k}and c

_{ψ}

_{min(j,l)}connects q

_{j}and q

_{l}. Recall that the endpoints of each segment on ϕ and ψ are two neighboring points of φ or ψ; |I − k| = 1 and |j − l| = 1. Hence, an edge in the graph shown in Figure 9 can only connect two nodes that are diagonally next to each other in the table and a node can be connected to, at most, four nodes by four edges. A pair of nodes can be connected to each other only by a single edge. The end nodes of (p

_{i}, q

_{j})-(p

_{k}, q

_{l}) are either the maxima or minima of these segments and their endpoints form horizontal line segments. Hence, from the Theorem 2, a pair of segments of ϕ and ψ represented by each edge in the graph is consistent with a one-parameter family of pairs of 3D curves that is symmetric with respect to the common symmetry plane Π

_{s}.

_{i}, q

_{j})-(p

_{k}, q

_{l}) and (p

_{i}, q

_{j})-(p

_{g}, q

_{h}) connected to a common node (p

_{i}, q

_{j}). These edges represent two pairs of segments of the 2D curves; a pair c

_{φ}

_{min(i,k)}and c

_{ψ}

_{min(j,l)}and a pair c

_{φ}

_{min(i,g)}and c

_{ψ}

_{min(j,h)}. The two segments c

_{φ}

_{min(i,k)}and c

_{φ}

_{min(i,g)}are connected to each other at their common endpoint p

_{i}(see Figure 8). In the same way, c

_{ψ}

_{min(j,l)}and c

_{ψ}

_{min(j,h)}are connected to each other at their common endpoint q

_{j}. Hence, these two pairs of 2D curves can be regarded as a pair of 2D curves whose endpoints are represented by (p

_{k}, q

_{l}) and (p

_{g}, q

_{h}) that are the end nodes of the path formed by the edges. Note that each (p

_{i}, q

_{j})-(p

_{k}, q

_{l}) and (p

_{i}, q

_{j})-(p

_{g}, q

_{h}) is consistent with infinitely many pairs of 3D continuous curves. Assume that they are symmetric with respect to the common symmetry plane Π

_{s}whose slant and depth are given. Then, two symmetric pairs of 3D continuous curves are uniquely determined; these 3D curves are backprojections of c

_{φ}

_{min(i,k)}, c

_{φ}

_{min(i,g)}, c

_{ψ}

_{min(j,l)}and c

_{ψ}

_{min(j,h)}, respectively. The 3D curves that are backprojections of c

_{φ}

_{min(i,k)}and c

_{φ}

_{min(i,g)}are connected to each other at their common endpoint that is a backprojection of p

_{i}; these two 3D curves can be regarded as a single 3D curve. The same way, the 3D curves that are backprojections of c

_{ψ}

_{min(j,l)}and c

_{ψ}

_{min(j,h)}can be regarded as a single 3D curve. It follows that the two symmetric pairs of 3D curves produced from c

_{φ}

_{min(i,k)}, c

_{φ}

_{min(i,g)}, c

_{ψ}

_{min(j,l)}and c

_{ψ}

_{min(j,h)}can be regarded as a single symmetric pair of 3D continuous curves. Their endpoints are backprojections of the 2D points represented by the end nodes (p

_{k}, q

_{l}) and (p

_{g}, q

_{h}) of the continuous path. This can be generalized to all segments of ϕ and ψ as follows. A continuous path of the edges in the table in Figure 9 represents a pair of 2D continuous curves; the 2D curves are composed of the 2D segments of φ and ψ and their endpoints are represented by the end nodes of the path. The 2D curves are consistent with an infinite number of symmetric pairs of 3D continuous curves. The endpoints of the 3D curves are backprojections of the points represented by the end nodes of the path. So, if there is a continuous path connecting (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) in the graph in Figure 9, then this path represents a pair of 2D continuous curves φ and ψ and this pair of the 2D curves is consistent with a one-parameter family of symmetric pairs of 3D continuous curves.

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) in the graph will now be proved by using concepts from a graph theory. A graph is called connected if there is a continuous path of edges between every pair of nodes in the graph. If (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) belong to a connected graph, there is a path connecting (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}). A connected graph has even number of nodes of odd degree, where degree of a node refers to number of edges connected to the node [20]. If (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) are the only nodes of odd degree in the graph, they must belong to the same connected graph and there must be a continuous path connecting (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}). Next, we provide a classification of possible nodes in the graph and show that there are only four types: of degree zero, one, two or four. This will conclude the proof.

_{1}, …, p

_{m}on φ and q

_{1}, …, q

_{n}on ψ. If a node (p

_{i}, q

_{j}) exists in the table, p

_{i}and q

_{j}form a horizontal line segment in Figure 8. Note that (p

_{i}, q

_{j}) can be connected to, at most, four nodes in the graph that are diagonally next to (p

_{i}, q

_{j}): (p

_{i}

_{−1}, q

_{j}

_{−1}), (p

_{i}

_{−1}, q

_{j}

_{+1}), (p

_{i}

_{+1}, q

_{j}

_{−1}) and (p

_{i}

_{+1}, q

_{j}

_{+1}). Hence, the maximum degree of each node in the table is four. These four neighboring nodes represent possible pairs of neighboring points of p

_{i}along φ and q

_{j}along ψ. Consider a pair of the neighboring points p

_{i}

_{+1}and q

_{j}

_{+ 1}. The node (p

_{i}

_{+1}, q

_{j}

_{+1}) exists if and only if p

_{i}

_{+1}and q

_{j}

_{+1}are a corresponding pair; they form a horizontal line segment in Figure 8. Note that p

_{i}

_{+ 1}and p

_{i}are connected by c

_{φi}and q

_{j}

_{+1}and q

_{j}are connected by c

_{ψj}. If both (p

_{i}

_{+1}, q

_{j}

_{+1}) and (p

_{i}, q

_{j}) exist in the graph, they are connected by (p

_{i}

_{+1}, q

_{j}

_{+1})-(p

_{i}, q

_{j}) representing a pair of segments c

_{φi}and c

_{ψj}. Therefore, the number of edges in the graph connected to (p

_{i}, q

_{j}) can be computed by verifying the existence of the four neighboring nodes.

_{1}, …, p

_{m}and q

_{1}, …, q

_{n}. First, these points can be classified into three types: local maxima, local minima and points at which the curve is monotonic (Figure 10). If a point is a local maximum, its neighboring points are lower than the local maximum. If a point is a local minimum, its neighboring points are higher than the local minimum. If a point is a “monotonic point”, one of its neighboring points is higher and the other is lower. Next, based on the classification of the points, the corresponding pairs of the points can be classified into three types (Figure 11). Type (i): if two monotonic points are corresponding, a node representing this pair of points is connected to two nodes by two edges. Hence, the degree of this node is two; Type (ii): if a monotonic point and a local maximum/minimum are corresponding, the degree of a node representing this pair is also two; Type (iii): if two local maxima/minima are corresponding, the degree of a node representing this pair is four; Type (iv): if a local maximum and a local minimum are corresponding, the degree of a node representing this pair is zero. From these facts, the degree of any node which does not represent the endpoints of the two curves is always even (0, 2 or 4).

_{0}, q

_{0}) and (p

_{m}

_{+ 1}, q

_{n}

_{+1}). Recall that p

_{0}= e

_{φ}

_{0}, p

_{m}

_{+1}= e

_{φ}

_{1}, q

_{0}= e

_{ψ}

_{0}and q

_{n}

_{+1}= e

_{ψ}

_{1}; so, the line segments p

_{0}q

_{0}and p

_{m}

_{+1}q

_{n}

_{+1}are horizontal in Figure 8. Hence, these pairs of the endpoints are corresponding pairs and the nodes representing these pairs exist in the graph. Note that these endpoints are global maxima and minima of φ and ψ; the global maxima form a corresponding pair and the global minima form a corresponding pair. Recall that if two local maxima or two local minima form a corresponding pair, there are four corresponding pairs of their neighboring points. However, each endpoint has only one neighboring point. Hence, there is one corresponding pair of the neighboring points for each pair of the endpoints. Therefore, each (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) is connected to a single node and their degrees are one, which is an odd number.

_{φ}

_{0}and e

_{ψ}

_{0}or e

_{φ}

_{1}and e

_{ψ}

_{1}is tangent to the curve of the extremum, is analogous to Type (ii) in Figure 11. The extremum of the curve corresponds to an endpoint of the other curve that is on the horizontal tangent line. Unlike Type (ii), the endpoint has only one neighboring point that forms a corresponding pair with the neighboring points of the local extremum. Hence, the degree of a node representing the pair of the local extremum and the endpoint is also two, which is an even number.

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) whose degrees are odd (1) and the degrees of all other nodes are even (0, 2 or 4). Hence, (p

_{0}, q

_{0}) and (p

_{m}

_{+1}, q

_{n}

_{+1}) must belong to the same connected graph and these two nodes are connected by a continuous path of edges. This continuous path in the graph represents the correspondences between 3D continuous curves of a symmetric pair whose orthographic projections are the 2D curves φ and ψ, respectively. Once the correspondences are formed, the pair of the 3D curves can be produced using Equations (15) and (16). Note that the slant of the symmetry plane Π

_{s}is a free parameter of the 3D symmetric interpretation of φ and ψ under an orthographic projection. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 8 is shown in Demo 6 in supplemental material [2].

**Theorem**

**4:**

_{φ}

_{0}and e

_{φ}

_{1}, and the endpoints of ψ be e

_{ψ}

_{0}and e

_{ψ}

_{1}. Let a line connecting e

_{φ}

_{0}and e

_{ψ}

_{0}be l

_{0}and that connecting e

_{φ}

_{1}and e

_{ψ}

_{1}be l

_{1}. Assume that φ and ψ have the following properties: (i) l

_{0}and l

_{1}intersect at a point v that is not on φ or ψ; (ii) l

_{0}and l

_{1}do not have any intersection with φ and ψ; (iii) v is not between e

_{φ}

_{0}and e

_{ψ}

_{0}or between e

_{φ}

_{1}and e

_{ψ}

_{1}; and (iv) a half line that emanates from v and intersects with φ has one or a finite number of intersections with ψ and vice versa. Then, there exists a pair of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space for a given center of projection F, such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is a perspective projection of Φ and ψ is a perspective projection of Ψ.

**Proof:**In the proof of Theorem 3 for the case of an orthographic projection, the 2D curves φ and ψ were divided by lines which were parallel to e

_{φ}

_{0}e

_{ψ}

_{0}and were tangent to either of the 2D curves. In the case of a perspective projection, φ and ψ are divided by lines which emanate from the vanishing point v and are tangent to either of the 2D curves. The rest of this proof is identical to that of Theorem 3. The only difference is that in the case of a perspective projection the 3D interpretation is unique—the slant of the symmetry plane is not a free parameter.

**Theorem**

**5:**

_{0}and l

_{1}which satisfy the following properties: (i) l

_{0}and/or l

_{1}is either tangent to both φ and ψ or passes through their endpoints, (ii) l

_{0}||l

_{1}, (iii) l

_{0}and l

_{1}do not have any intersection with φ and ψ and (iv) a line that is parallel to l

_{0}and intersects φ has one or a finite number of intersections with ψ and vice versa (see Figure 12). Then, there exist infinitely many pairs of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space, such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is an orthographic projection of Φ and ψ is an orthographic projection of Ψ.

_{s}. Their orthographic projections in the 2D image coincide with φ′ and ψ′, as well as with φ and ψ.

_{0}and l

_{1}be horizontal. In this case, tangent points of φ and ψ to l

_{0}or l

_{1}are either global maxima or minima of φ and ψ along a vertical axis on the 2D image. Assume that the endpoints of these curves are not global extrema—the case when they are global extrema has been considered in the previous theorems. Let the tangent points of φ to l

_{0}and l

_{1}be t

_{φ}

_{0}and t

_{φ}

_{1}. Let an endpoint of φ, which is closer (as measured by the arc length along the curve) to t

_{φ}

_{0}be e

_{φ}

_{0}, and that, which is closer to t

_{φ}

_{1}be e

_{φ}

_{1}. The same way, let the tangent points of ψ to l

_{0}or l

_{1}be t

_{ψ}

_{0}and t

_{ψ}

_{1}, and the endpoints ψ be e

_{ψ}

_{0}and e

_{ψ}

_{1}.

_{φ}

_{0}, e

_{φ}

_{1}, e

_{ψ}

_{0}and e

_{ψ}

_{1}, which are endpoints of φ and ψ. The extension starting from e

_{φ}

_{0}is identical to the segment of φ between e

_{φ}

_{0}and t

_{φ}

_{0}. Similarly, the extension starting from e

_{φ}

_{1}is identical to the segment of φ between e

_{φ}

_{1}and t

_{φ}

_{1}. Let this new curve be φ′. Let the endpoints of φ′ be e′

_{φ}

_{0}and e′

_{φ}

_{1}, so that the positions of e′

_{φ}

_{0}and e′

_{φ}

_{1}are respectively the same as those of t

_{φ}

_{0}and t

_{φ}

_{1}in the 2D image. The same way, let the extended curve produced from ψ be ψ′ and the endpoints of ψ′ be e′

_{ψ}

_{0}and e′

_{ψ}

_{1}. The positions of e′

_{ψ}

_{0}and e′

_{ψ}

_{1}are respectively the same as those of t

_{ψ}

_{0}and t

_{ψ}

_{1}in the 2D image. The curves φ′ and ψ′ perfectly overlap φ and ψ in the 2D image (Figure 12). Therefore, the 3D interpretations of φ′ and ψ′ are also consistent with the 3D interpretations of φ and ψ. It is easy to see that φ′ and ψ′ satisfy the assumptions of Theorem 3. Therefore, it follows from the proof of Theorem 3 that there exists a one-parameter family of symmetric pairs of continuous 3D curves Φ′ and Ψ′, such that φ′ is an orthographic projection of Φ′ and ψ′ is an orthographic projection of Ψ′. This, in turn, implies that φ and ψ are also orthographic projection of Φ′ and Ψ′. A symmetric pair of 3D curves produced from the pair of 2D curves in Figure 12 is shown in Demo 7 in supplemental material [2] (It looks like each of the 3D curves in Demo 6 has four endpoints, rather than two, as would be expected from Theorem 3. It also looks like each of the 3D curves has two bifurcations. All of the four points that look like endpoints are actually 180° turns of the 3D curve. The real endpoints of the 3D curve are at the bifurcations).

**Proof:**

**Figure 12.**φ and ψ (red solid curves) are 2D curves. e

_{φ}

_{0}and e

_{φ}

_{1}are the endpoints of φ, and e

_{ψ}

_{0}and e

_{ψ}

_{1}are the end points of ψ. l

_{0}is tangent to both φ and ψ at t

_{φ}

_{0}and t

_{ψ}

_{0}, and l

_{1}is tangent to both φ and ψ at t

_{φ}

_{1}and t

_{ψ}

_{1}. l

_{0}and l

_{1}are parallel to the x-axis and do not have any intersection with φ and ψ. The two curves, φ and ψ are extended by adding arcs (blue dotted curves) that start from e

_{φ}

_{0}, e

_{φ}

_{1}, e

_{ψ}

_{0}and e

_{ψ}

_{1}. The additional arcs are identical to the segments of φ and ψ (note that the blue dotted curves are perfectly overlapping the red solid curves in the image). They end at e′

_{φ}

_{0}, e′

_{φ}

_{1}, e′

_{ψ}

_{0}and e′

_{ψ}

_{1}whose positions are respectively the same as those of t

_{φ}

_{0}, t

_{φ}

_{1}, t

_{ψ}

_{0}and t

_{ψ}

_{1}. See Demo 7 in supplemental material for an interactive illustration of the 3D symmetric curves produced from φ and ψ [2].

**Theorem**

**6:**

_{0}and l

_{1}which satisfy the following properties: (i) l

_{0}and/or l

_{1}is either tangent to both φ and ψ or passes through their endpoints; (ii) l

_{0}and l

_{1}intersect at a point v that is not on φ or ψ; (iii) l

_{0}and l

_{1}do not have any other intersections with φ and ψ; (iv) v is not between e

_{φ}

_{0}and e

_{ψ}

_{0}or between e

_{φ}

_{1}and e

_{ψ}

_{1}, and (v) a half line that emanates from v and intersects with φ has one or a finite number of intersections with ψ and vice versa. Then, there exists a pair of continuous curves Φ and Ψ and a plane Π

_{s}in a 3D space for a given center of projection F, such that Φ and Ψ are mirror-symmetric with respect to Π

_{s}and that φ is a perspective projection of Φ and ψ is a perspective projection of Ψ.

## 3. Discussion

**Figure 13.**A 2D closed curve and its two different symmetric interpretations. (

**a**) An orthographic projection of a closed symmetric curve. Each of the two symmetric halves of the curve is planar. The corresponding pairs of the points of the curve form horizontal line segments; (

**b**) The front view of the symmetric interpretation of the curve in (

**a**) using the “correct” correspondence. The slant and the tilt of the symmetry plane of the symmetric interpretation were set to 40° and 0°, respectively. Each of the two symmetric halves of the 3D curve is planar; (

**c**) The front view of the symmetric interpretation of the curve in (

**a**) using the “wrong” correspondence. The slant and the tilt of the symmetry plane were set to 40° and −25°, respectively. Neither of the two symmetric halves of the 3D curve is planar. See Demo 8 in supplemental material for an interactive illustration showing these two different 3D curves produced from the same 2D image [2].

## Acknowledgements

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**Figure 1.**A 2D closed curve that looks like a 2D projection of a 3D mirror-symmetric curve. See Demo 1 in supplemental material for an interactive illustration of the 3D symmetric interpretation produced from this 2D curve [2].

**Figure 2.**An asymmetric pair of 2D curves and the 3D symmetric interpretation. (

**a**) An asymmetric pair of 2D curves. These curves can be interpreted as a 2D perspective projection of a symmetric pair of 3D curves whose symmetry plane is slanted at an angle of 40° (see Theorem 1); (

**b**) Three different views of the 3D symmetric interpretation produced from the pair of the 2D curves in (

**a**). The numbers in the bottom are the values of the slant σ

_{s}of the symmetry plane of the symmetric pair of the 3D curves. For σ

_{s}equal to 40°, the image is identical to that in (

**a**). When σ

_{s}is 90°, its 2D projection itself becomes symmetric. See Demo 2 in supplemental material for an interactive illustration of the 3D symmetric curves [2].

**Figure 4.**Front and side views of a symmetric pair of 3D curves produced from the pair of 2D curves in Figure 3. The distance z

_{f}between the center of projection and the image plane, together with the vanishing point v determine the slant σ

_{s}of the symmetry plane Π

_{s}(the slant is 30° in this case). See Demo 3 in supplemental material for an interactive illustration of the 3D symmetric curves [2]. Note that a perspective projection is used in Demo 3. As a result, the two curves in the side view do not project to the same curve on the image: the farther curve projects to a smaller image. The side view in this figure was computed using an orthographic projection. As a result, the two symmetric 3D curves project to the same 2D image.

**Figure 9.**This table represents pairs of segments and pairs of points on of ϕ and ψ. The rows represent the points p

_{0}, …, p

_{m}

_{+1}on φ. The columns represent the points q

_{0}, …, q

_{n+}

_{1}on ψ. A circle at (p

_{i}, q

_{j}), which is a node in the graph, represents a corresponding pair of points p

_{i}and q

_{j}; p

_{i}and q

_{j}form a horizontal line segment in Figure 8. The top left and bottom right nodes represent pairs of endpoints of φ and ψ. An edge (p

_{i}, q

_{j})-(p

_{k}, q

_{l}) (or (p

_{k}, q

_{l})-(p

_{i}, q

_{j})) connecting (p

_{i}, q

_{j}) and (p

_{k}, q

_{l}) represents a pair of segments of ϕ and ψ.

**Figure 10.**Three types of points (solid circles) on a 2D curve and their neighboring points (open circles).

**Figure 11.**Four types of pairs of points (solid circles) and their neighboring points (open circles). See text for more information.

© 2011 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sawada, T.; Li, Y.; Pizlo, Z.
Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation. *Symmetry* **2011**, *3*, 365-388.
https://doi.org/10.3390/sym3020365

**AMA Style**

Sawada T, Li Y, Pizlo Z.
Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation. *Symmetry*. 2011; 3(2):365-388.
https://doi.org/10.3390/sym3020365

**Chicago/Turabian Style**

Sawada, Tadamasa, Yunfeng Li, and Zygmunt Pizlo.
2011. "Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation" *Symmetry* 3, no. 2: 365-388.
https://doi.org/10.3390/sym3020365