SHAPE DEPENDENT MOTION INTERPOLANTS FOR PLANAR OBJECTS

Abstract

Kinematics is most commonly about the motion of unbounded spaces. This paper deals with the kinematics of bounded shapes in a plane. This paper studies the problem of motion interpolation of a planar object with its shape taken into consideration. It applies and extends a shape dependent distance measure between two positions in the context of motion interpolation. Instead of using a fixed reference frame, a shape-dependent inertia frame of reference is used for formulating the distance between positions of a rigid object in a plane. The resulting distance function is then decomposed in two orthogonal directions and is used to formulate an interpolating function for the distance functions in these two directions. This leads to a shape dependent interpolation of translational components of a planar motion. In difference to the original concept of Kazerounian and Rastegar that comes with a shape dependent measure of the angular motion, it is assumed in this paper that the angular motion is shape independent as the angular metric is dimensionless. The resulting distance measure is not only a combination of translation and rotation parameters but also depends on the area moments of inertia of the object. It derives the explicit expressions for decomposing the shape dependent distance in two orthogonal directions, which is then used to obtain shape dependent motion interpolants in these directions. The resulting interpolants have similarities to the well-known spherical linear interpolants widely used in computer graphics in that they are defined using sinusoidal functions instead of linear interpolation in Euclidean space. The path of the interpolating motion can be adjusted by different choice of shape parameters. Examples are provided to illustrate the effect of object shapes on the resulting interpolating motions.

1. Introduction

Kinematics is concerned with the study of motion of objects in space without taking into account the forces that cause the motion. In Theoretical Kinematics [1], it is concerned with motion of unbounded infinite spaces which contain idealized geometric elements such as points, lines, and planes and studies rigid transformations in Euclidean spaces such that all distances among these geometric elements remain fixed. As everything that moves has kinematic aspects, kinematics has applications in many fields such as mechanisms, robotics, computer graphics and animation. In such applied kinematics, it is often not sufficient to consider a moving object as an unbounded space, and some aspects of its shape and size has to be taken into account. In the theory of mechanisms and machines, the classical branch of applied kinematics, it is not the shapes and size of the moving objects themselves, but rather the shapes and dimensions of kinematic constraints provided by mechanical contrivances, that determine the properties of the output motion. This paper, however, deals with a different type of kinematics, that of bounded objects, where not only rigid-body transformations of unbounded spaces but also object shape and size contribute towards the combined outcome. One well-known example is swept volume analysis, where the outcome is a combination of a rigid-body motion and the shape of an object undergoing the motion. Swept volume analysis plays a key role in NC tool motion simulation, as well as collision avoidance in robot motion planning. Motion animation in computer graphics is another example, where it is concerned with the visual representation of the deformation and movement of objects with bounded shapes and sizes.

Kazerounian and Rastegar [2, 3] were the first to propose the concept of shape dependent object norms for quantifying the “distance” between two positions of a rigid object in plane and spatial kinematics. The distance function is defined as the weighted average of all distances between all corresponding points of an object at two different positions. The resulting distance function is explicitly expressed in terms of the moment of inertia of the object measured in the fixed coordinate frame and thus shape dependent. In addition, it is independent of the choice of both coordinate frames. Using this novel concept, Rastegar et al. [4] proposed an object shape dependent kinematic manipulability measure for path and trajectory synthesis and shape optimization.

In this paper, the concept of shape dependent distance measure for planar displacements is adapted to the development of a planar motion interpolant that is shape dependent or shape modulated. Instead of using a fixed reference frame, a shape-dependent inertia frame of reference is used for formulating the distance between positions of a rigid object in a plane. The resulting distance function is then decomposed in two orthogonal directions and is used to formulate an interpolating function for the distance functions in these two directions. This leads to a shape dependent interpolation of translational components of a planar motion. In difference to the original concept of Kazerounian and Rastegar that comes with a shape dependent measure of the angular motion, it is assumed in this paper that the angular motion is shape independent as the angular metric is dimensionless.

The paper is organized as follows. Section 2 introduces shape dependent distance and its components measured in the principal coordinate frame (PCF). Section 3 develops shape dependent motion interpolants associated with PCF. Section 4 expanded the scope to the general coordinate frames. Section 5 presents an example to illustrate the effect of the object shape on the interpolating motion.

2. Shape Dependent Distance Measured in the Principal Coordinate Frame

Consider a simply connected planar object with an arbitrary shape (Figure 1). The area moment of inertia, also known as the second moment of area, is a geometric property of a planar shape. It can be expressed in tensor form by the following real symmetric matrix:

$$\left[\text{I}\right]=\left[\begin{array}{ll}{I}_{xx}\hfill & I_{xy}\hfill \\ I_{xy}\hfill & I_{yy}\hfill \end{array}\right]$$ (1)

where diagonal elements are the area moments of inertia with respect to $x$ and $y$ axes and off-diagonal elements are the products of inertia. After eigendecomposition, the inertia matrix (1) yields principal areas of inertia as well as principal axes (or directions) that define an ellipse in the principal coordinate frame (PCF)

FIGURE 1.

A planar object and its inertia ellipse.

In [5], Venkataramanujam and Larochelle adapted the notion of PCF to a set of planar displacements to set up a fixed reference frame that leads to a distance measure with invariant properties. In this paper, PCF is used as a body-fixed or moving coordinate frame and we denote $U$-axis and $V$-axis as the major and minor axis respectively. The principal moments of inertia, where $I_u\le I_v$, are used as shape parameters, and it will be shown that the associated ellipse captures the influence of the object shape on a shape dependent distance measure between two positions of an bounded object.

Now consider two positions of an object whose shape is represented by an ellipse as is shown in Figure 2. Let $O_1{U}_1{V}_1$ and $O_2{U}_2{V}_2$ be the principal coordinate frames of the object at its initial and final positions. The relative displacement of ${\text{O}}_2{U}_2{V}_2$ with respect to $O_1{U}_1{V}_1$ is defined by the vector $\text{d}=\left(d_u,d_v\right)$ connecting the two origins and the angle $\theta$ between the axes $U_1$ and $U_2$. Furthermore, let ${\text{p}}_1\left(u_1,v_1\right)$ and ${\text{p}}_2\left(u_2,v_2\right)$ denote vectors representing the positions of an arbitrary point of the object at $O_1{U}_1{V}_1$ and $O_2{U}_2{V}_2$ respectively, both measured in $O_1{U}_1{V}_1$. Then we have

$$\left(\begin{array}{c}{u}_2\\ v_2\end{array}\right)=\left[\begin{array}{cc}\text{cos}\theta & -\text{sin}\theta \\ \text{sin}\theta & \text{cos}\theta \end{array}\right]\left(\begin{array}{c}{u}_1\\ v_1\end{array}\right)+\left(\begin{array}{c}{d}_u\\ d_v\end{array}\right).$$ (2)

The displacement from ${\text{p}}_1$ to ${\text{p}}_2$ is obtained as

$$\left(\begin{array}{c}\Delta u\\ \Delta v\end{array}\right)=\left(\begin{array}{c}{u}_2-u_1\\ v_2-v_1\end{array}\right)=\left(\begin{array}{c}{u}_1\left(\text{cos}\theta -1\right)-v_1\text{sin}\theta +d_u\\ u_1\text{sin}\theta +v_1\left(\text{cos}\theta -1\right)+d_v\end{array}\right)$$ (3)

FIGURE 2.

Two positions of a planar object with its PCF. Note that the object shape is not limited to ellipse and can be of any simply connected shape.

We follow the notion of object norms proposed in [2–4] and define a shape dependent distance between two objects as the average distance between all points of the object at the two given positions:

$$D^2=\frac{1}{S}{\iint }_S\left(\Delta u^2+\Delta v^2\right)dS$$ (4)

where $dS$ is an infinitesimal element of area and $S$ is the total area of the object. Substitute $\Delta u$ and $\Delta v$ in (3) into (4) to obtain:

$$D^2=\frac{1}{S}\left\{{\iint }_{S}2\left(1-\text{cos}\theta \right)u_1^{2}dS+{\iint }_{S}2\left(1-\text{cos}\theta \right)v_1^{2}dS-{\iint }_{S}2\left[d_u\left(1-\text{cos}\theta \right)-d_v\text{sin}\theta \right]u_{1}dS-{\iint }_{S}2\left[d_v\left(1-\text{cos}\theta \right)+d_u\text{sin}\theta \right]v_{1}dS+{\iint }_S\left(d_u^2+d_v^2\right)dS\right\}.$$ (5)

Introducing the centroid and area moments of inertia, the above equation can be further simplified as

$$D^2=\frac{1}{S}\left\{2\left(1-\text{cos}\theta \right)I_v+2\left(1-\text{cos}\theta \right)I_u-2\left[d_u\left(1-\text{cos}\theta \right)-d_v\text{sin}\theta \right]{\overline{u}}_{1}S-2\left[d_v\left(1-\text{cos}\theta \right)+d_u\text{sin}\theta \right]{\overline{v}}_{1}S+\left(d_u^2+d_v^2\right)S\right\}$$ (6)

where $I_u,I_v\left(I_u<I_v\right)$ denote the principal area-moments of inertia, $\left({\overline{u}}_1,{\overline{v}}_1\right)$ is the centroid of the object, which has the coordinates (0, 0) in $O_1{U}_1{V}_1$. The last term $d_u^2+d_v^2=d^2$ where $d$ is the distance between $O_1$ and $O_2$ (Figure 2).

Introducing the radii of gyration, $k_u$ and $k_v\left(k_u\le k_v\right)$, such that

$$I_u=k_u^{2}S,I_v=k_v^{2}S,$$ (7)

we obtain the following form for the shape dependent distance $D$ between two positions of the object

$$D^2=d^2+4{\text{sin}}^2\frac{\theta }{2}\left(k_u^2+k_v^2\right).$$ (8)

Let $a$ and $b$ be the length of semi-major and semi-minor axis of the inertia ellipse respectively. The radii of gyration for the ellipse can be expressed as

$$k_u=\frac{b}{2},k_v=\frac{a}{2}.$$ (9)

Substituting Eqs.(9) into Eq. (8), the distance measure is rewritten as

$$D^2=d^2+\left(a^2+b^2\right){\text{sin}}^2\frac{\theta }{2}.$$ (10)

Therefore, the shape dependent distance $D$ is a combination of the translational distance $d$ and the rotation angle $\theta$ but weighted with the shape parameter $\sqrt{a^2+b^2}$, which, of course, indicates the size of the object.

There have been many attempts to define distance measures that combine translation and rotation and study the issue of coordinate-frame invariance ([6–15]). One of the key considerations in all these work is on how to add a weight factor to the rotation angle so that it takes on a length scale. This includes the use of spherical displacements to approximate planar displacements [12, 14] and by extension hyperspherical displacements to approximate spatial displacements [13], as well as the notion of characteristic length [16]. More recently, it has been shown in Ge et al. [17, 18] that the weight factor does not play a role in computing the optimal average displacement based on a least squares criterion. All these work, however, are restricted to rigid-body kinematics of unbounded spaces and thus did not take into account the shape of the object when combining rotation with translation.

The main purpose of this paper, however, is to develop motion interpolants that incorporate the shape and size of an object. To this end, we seek to decompose the shape dependent distance $D$ into two components along two orthogonal directions $U_1$ and $V_1$. In view of (4), we separate $D^2$ into $D_u^2$ and $D_v^2$ such that

$$D^2=D_u^2+D_v^2,$$ (11)

where

$$D_u^2=\frac{1}{S}{\iint }_S\Delta u^{2}dS,D_v^2=\frac{1}{S}{\iint }_S\Delta v^{2}dS.$$ (12)

Following a procedure similar to that from (5) to (10), we obtain

$$\begin{gathered} D_u^2=d_u^2+\left(b^2{\text{cos}}^2\frac{\theta }{2}+a^2{\text{sin}}^2\frac{\theta }{2}\right){\text{sin}}^2\frac{\theta }{2}, \\ {D}_v^2=d_v^2+\left(b^2{\text{sin}}^2\frac{\theta }{2}+a^2{\text{cos}}^2\frac{\theta }{2}\right){\text{sin}}^2\frac{\theta }{2}. \end{gathered}$$ (13)

In [4], a shape dependent angular measure is also proposed. For the purpose of motion interpolation, however, we view the rotation angle as a property of the moving space and thus shape independent.

3. Interpolating Shape Dependent Distances

In this section, we seek to develop interpolants for the shape dependent distance $D$ as well as their components $D_u$ and $D_v$.

3.1. Interpolating the Distance $D$

Now consider the problem of interpolating the shape dependent distance $D$ as given by (10). Let $\left({\text{d}}_{\text{m}},{\theta }_m\right)$ represent an interpolated position between $O_1{U}_1{V}_1$ and $O_2{U}_2{V}_2$ and $D_m$ the shape dependent distance from $O_1{U}_1{V}_1$ to the interpolated position. We seek to determine an interpolating function $f\left(t\right)\left(0\le t\le 1\right)$ such that

$$D_m\left(t\right)=f\left(t\right)D=f\left(t\right)\sqrt{d^2+\left(a^2+b^2\right){\text{sin}}^2\frac{\theta }{2},}$$ (14)

and $f\left(0\right)=0$ and $f\left(1\right)=1$.

We first apply a linear interpolation to the rotation angle, i.e., ${\theta }_m=t\theta$. Then it follows from (10) that

$$D_m^2\left(t\right)=d_m^2\left(t\right)+\left(a^2+b^2\right){\text{sin}}^2\frac{t\theta }{2},$$ (15)

where $d_m^2\left(t\right)$ is the unknown interpolating function for the translation component. The substitution of (14) into (15) yields

$$d_m^2\left(t\right)+\left(a^2+b^2\right){\text{sin}}^2\frac{t\theta }{2}=f{(t)}^2{d}^2+f{(t)}^2\left(a^2+b^2\right){\text{sin}}^2\frac{\theta }{2}.$$ (16)

Equating the corresponding terms, we obtain

$$d_m\left(t\right)=f\left(t\right)d,$$ (17)

and

$$f\left(t\right)=\frac{\text{sin}\frac{t\theta }{2}}{\text{sin}\frac{\theta }{2}}.$$ (18)

Therefore, the same interpolating function $f\left(t\right)$ works for both $d_m\left(t\right)$ and $D_m\left(t\right)$. It is an interesting coincidence that $f\left(t\right)$ happens to be one of the two spherical linear interpolants for quaternion interpolation in [19].

3.2. Interpolating $D_u$ and $D_v$

Let ${\theta }_m=t\theta$ as before and let ${\text{d}}_m=\left(d_{mu},d_{mv}\right)$ be the coordinates of the interpolated position. Let the components of distance from the interpolated position to $O_1{U}_1{V}_1$ in the $U_1$ and $V_1$ directions be denoted as $D_{mu},D_{mv}$, respectively. By following the same steps as in Section 3.1, we can show that

$$D_{mu}\left(t\right)=f_u\left(t\right)D_u,D_{mv}\left(t\right)=f_v\left(t\right)D_v$$ (19)

$$d_{mu}\left(t\right)=f_u\left(t\right)d_u,d_{mv}\left(t\right)=f_v\left(t\right)d_v$$ (20)

where

$$\begin{gathered} f_u\left(t\right)=\frac{\text{sin}\frac{t\theta }{2}\sqrt{b^2{\text{cos}}^2\frac{t\theta }{2}+a^2{\text{sin}}^2\frac{t\theta }{2}}}{\text{sin}\frac{\theta }{2}\sqrt{b^2{\text{cos}}^2\frac{\theta }{2}+a^2{\text{sin}}^2\frac{\theta }{2}}}, \\ {f}_v\left(t\right)=\frac{\text{sin}\frac{t\theta }{2}\sqrt{b^2{\text{sin}}^2\frac{t\theta }{2}+a^2{\text{cos}}^2\frac{t\theta }{2}}}{\text{sin}\frac{\theta }{2}\sqrt{b^2{\text{sin}}^2\frac{\theta }{2}+a^2{\text{cos}}^2\frac{\theta }{2}}}. \end{gathered}$$ (21)

Once again, the same interpolating functions work for both shape dependent distance and translation distance.

Note that the shape parameters $a,b$ appear in the motion interpolants $f_u\left(t\right)$ and $f_v\left(t\right)$. They represent the effect of object shape and size on the interpolating motion. It is the ratio $e=b/a$ that matters, not the size of the object. In the special case when $a=b$, the interpolants $f_u\left(t\right),f_v\left(t\right)$ are no longer shape dependent but are given by

$$f_u\left(t\right)=f_v\left(t\right)=\frac{\text{sin}\frac{t\theta }{2}}{\text{sin}\frac{\theta }{2}}.$$ (22)

4. Motion Interpolation in General Coordinate Frames

This section derives the expressions for the shape dependent motion interpolants for a general set of moving and fixed reference frames.

4.1. Shape Dependent Distance and Its Components

Consider two positions of an object represented by a moving frame $xy$ with respect to a fixed frame $XY$ as shown in Figure 3. Let ${\text{T}}_i=\left({\text{d}}_i,{\theta }_i\right)$ where ${\text{d}}_i=\left(d_{ix},d_{iy}\right)\left(i=1,2\right)$ represent two positions of the moving frame, and let $\overline{\text{r}}=\left({\overline{r}}_x,{\overline{r}}_y,\varphi \right)$ represent the position and orientation of PCF (or $UV$ frame) of the object with respect to the moving frame $xy$. Furthermore, we let $\text{r}\left(r_x,r_y\right)$ represent the coordinate vector of a point of the object measured relative to the moving frame $xy$, and let ${\text{p}}_1\left(x_1,y_1\right)$ and ${\text{p}}_2\left(x_2,y_2\right)$ denote the coordinate vectors of the same point at two different positions but measured with respect to the fixed coordinate frame $XY$. Then we have the following coordinate transformations:

$$\left(\begin{array}{c}{x}_1\\ y_1\end{array}\right)=\left[\begin{array}{cc}\text{cos}{\theta }_1& -\text{sin}{\theta }_1\\ \text{sin}{\theta }_1& \text{cos}{\theta }_1\end{array}\right]\left(\begin{array}{c}{r}_x\\ r_y\end{array}\right)+\left(\begin{array}{c}{d}_{1x}\\ d_{1y}\end{array}\right),$$ (23)

and

$$\left(\begin{array}{c}{x}_2\\ y_2\end{array}\right)=\left[\begin{array}{cc}\text{cos}{\theta }_2& -\text{sin}{\theta }_2\\ \text{sin}{\theta }_2& \text{cos}{\theta }_2\end{array}\right]\left(\begin{array}{c}{r}_x\\ r_y\end{array}\right)+\left(\begin{array}{c}{d}_{2x}\\ d_{2y}\end{array}\right).$$ (24)

FIGURE 3.

Planar Movement

Subtracting (23) from (24) to obtain $\Delta x=x_2-x_1,\Delta y=y_2-y_1$ in $X$ and $Y$ directions. Following a similar derivation from (4) to (10) and then introducing area moments of inertia and the centroid, we obtain shape dependent distance in its most general form:

$$D_{12}^2=\Delta d_x^2+\Delta d_y^2+\left(a^2+b^2\right){\text{sin}}^2\frac{{\theta }_{12}}{2}+4\text{sin}\frac{{\theta }_{12}}{2}\left[\Delta d_y\left(\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)-\Delta d_x\left(\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]+4{\text{sin}}^2\frac{{\theta }_{12}}{2}\left({\overline{r}}_x^2+{\overline{r}}_y^2\right),$$ (25)

where $\Delta d_x=d_{2x}-d_{1x},\Delta d_y=d_{2y}-d_{1y}$, and ${\theta }_{12}={\theta }_2-{\theta }_1$. The shape parameters $a$ and $b$ are the lengths of semi-major and semi-minor axis of the inertia ellipse respectively. In addition, the angle $\varphi$, which is the orientation of PCF with respect to the moving frame, does not contribute to the distance. Furthermore, when ${\overline{r}}_x$ and ${\overline{r}}_y$ are set to zero, which means that the moving coordinate frame is at the centroid of the object, Eq.(25) is reduced to

$$D_{12}^2=\Delta d_x^2+\Delta d_y^2+\left(a^2+b^2\right){\text{sin}}^2\frac{{\theta }_{12}}{2}$$ (26)

which is the same as Eq.(10). In this case, the choice of the fixed frame has no effect on the shape dependent distance.

It can be shown that $D_{12}^2$ can be decomposed into two components in the X and Y directions, i.e., $D_{12}^2=D_{12x}^2+D_{12y}^2$ where

$$D_{12x}^2={\left[\Delta d_x-2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]}^2+{\text{sin}}^2\frac{{\theta }_{12}}{2}\left[b^2{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)+a^2{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)\right],$$ (27)

$$D_{12y}^2={\left[\Delta d_y+2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]}^2+{\text{sin}}^2\frac{{\theta }_{12}}{2}\left[b^2{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)+a^2{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)\right].$$ (28)

Detailed derivations are given in Appendix A.

Each of $D_{12x}$ and $D_{12y}$ is a combination of three parts: the displacement parameters, the position and orientation of the object relative to the moving frame, as well as the lengths of two semi axes of the inertia ellipse for the object.

4.2. Motion Interpolation

Given two positions of an object $\left({\text{d}}_1,{\theta }_1\right)$ and $\left({\text{d}}_2,{\theta }_2\right)$ and denote the interpolated position as $\left({\text{d}}_{\text{m}},{\theta }_m\right)$, we can find an appropriate interpolation function that involves shape parameters of the object.

Based on the distance components in (27)(28), the components of distance between initial and interpolated positions are

$$D_{1mx}^2\left(t\right)={\left[\left(d_{mx}\left(t\right)-d_{1x}\right)-2\text{sin}\frac{{\theta }_{1m}\left(t\right)}{2}\left(\text{sin}\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}{\overline{r}}_y\right)\right]}^2+{\text{sin}}^2\frac{{\theta }_{1m}\left(t\right)}{2}\left[b^2{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}+\varphi \right)+a^2{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}+\varphi \right)\right],$$ (29)

$$D_{1my}^2\left(t\right)={\left[\left(d_{my}\left(t\right)-d_{1y}\right)+2\text{sin}\frac{{\theta }_{1m}\left(t\right)}{2}\left(\text{cos}\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}{\overline{r}}_y\right)\right]}^2+{\text{sin}}^2\frac{{\theta }_{1m}\left(t\right)}{2}\left[b^2{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}+\varphi \right)+a^2{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_m\left(t\right)}{2}+\varphi \right)\right].$$ (30)

As before, a linear interpolation of the rotation angle is used:

$${\theta }_m\left(t\right)={\theta }_1+t\left({\theta }_2-{\theta }_1\right).$$ (31)

Once again, we would like to determine the interpolating functions $f_x\left(t\right)$ and $f_y\left(t\right)$ such that

$$D_{1mx}\left(t\right)=f_x\left(t\right)D_{12x},D_{1my}\left(t\right)=f_y\left(t\right)D_{12y},$$ (32)

with the same end conditions $f_x\left(0\right)=f_y\left(0\right)=0$ and $f_x\left(1\right)=f_y\left(1\right)=1$. Following the same procedure as described from (14) to (18), we can show that

$$\begin{gathered} f_x\left(t\right)=\frac{\text{sin}\frac{t{\theta }_{12}}{2}\sqrt{b^2{\text{cos}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)+a^2{\text{sin}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)}}{\text{sin}\frac{{\theta }_{12}}{2}\sqrt{b^2{\text{cos}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)+a^2{\text{sin}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)}}, \\ {f}_y\left(t\right)=\frac{\text{sin}\frac{t{\theta }_{12}}{2}\sqrt{b^2{\text{sin}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)+a^2{\text{cos}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)}}{\text{sin}\frac{{\theta }_{12}}{2}\sqrt{b^2{\text{sin}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)+a^2{\text{cos}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)}}. \end{gathered}$$ (33)

Let $e=b/a\left(e\le 1\right)$ be the ratio of the lengths of two semi axes of the inertia ellipse. Then (33) becomes

$$\begin{gathered} f_x(t)=\frac{\text{sin}\frac{t{\theta }_{12}}{2}\sqrt{e^2{\text{cos}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)+{\text{sin}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)}}{\text{sin}\frac{{\theta }_{12}}{2}\sqrt{e^2{\text{cos}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)+{\text{sin}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)}} \\ {f}_y(t)=\frac{\text{sin}\frac{t{\theta }_{12}}{2}\sqrt{e^2{\text{sin}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)+{\text{cos}}^2\left({\theta }_1+\frac{t{\theta }_{12}}{2}+\varphi \right)}}{\text{sin}\frac{{\theta }_{12}}{2}\sqrt{e^2{\text{sin}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)+{\text{cos}}^2\left({\theta }_1+\frac{{\theta }_{12}}{2}+\varphi \right)}} \end{gathered}$$ (34)

Now both the orientation ${\theta }_1$ of the initial position and the orientation $\varphi$ of PCF with respect to the moving coordinate frame play a role in the interpolants, in addition to the shape parameters $e$. When $e=1$, i.e., when the ellipse reduces to a circle, we have

$$f_x\left(t\right)=f_y\left(t\right)=\frac{\text{sin}\frac{t{\theta }_{12}}{2}}{\text{sin}\frac{{\theta }_{12}}{2}}.$$ (35)

The interpolated position $\left(d_{mx}\left(t\right),d_{my}\left(t\right)\right)$ can also be solved from (32) as

$$d_{mx}\left(t\right)=d_{1x}+2\text{sin}\frac{t{\theta }_{12}}{2}\left(\text{sin}\left({\theta }_1+\frac{t{\theta }_{12}}{2}\right){\overline{r}}_x+\text{cos}\left({\theta }_1+\frac{t{\theta }_{12}}{2}\right){\overline{r}}_y\right)+f_x\left(t\right)\left[\left(d_{2x}-d_{1x}\right)-2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right],$$ (36)

$$d_{my}\left(t\right)=d_{1y}-2\text{sin}\frac{t{\theta }_{12}}{2}\left(\text{cos}\left({\theta }_1+\frac{t{\theta }_{12}}{2}\right){\overline{r}}_x-\text{sin}\left({\theta }_1+\frac{t{\theta }_{12}}{2}\right){\overline{r}}_y\right)+f_y\left(t\right)\left[\left(d_{2y}-d_{1y}\right)+2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right].$$ (37)

This above interpolants describe the motion of the moving coordinate frame $xy$ from one position to another that expressed in the fixed frame $XY$. If we would like to determine the motion of the object, there are two approaches: One is based on the motion of the moving coordinate frame $\left(d_{mx},d_{my},{\theta }_m\right)$, we can carry out transformation between PCF and the moving frame, the other is using the displacement of PCF with respect to the fixed frame directly and following the same procedure to get the motion of the object. The detail of the first approach is shown in Appendix B. The second approach is presented as follows:

Let ${\text{T}}_{C1}=\left(C_{1x},C_{1y},{\theta }_{C1}\right)$ and ${\text{T}}_{C2}=\left(C_{2x},C_{2y},{\theta }_{C2}\right)$ be the displacements of PCF with respect to the fixed coordinate frame.

In this case, we have

$$\begin{gathered} {\theta }_{C1}={\theta }_1+\varphi ,{\theta }_{C2}={\theta }_2+\varphi , \\ {\theta }_{C12}={\theta }_{C2}-{\theta }_{C1}={\theta }_2-{\theta }_1={\theta }_{12}. \end{gathered}$$ (38)

It follows from (34) that

$$\begin{gathered} f_x(t)=\frac{\text{sin}\frac{t{\theta }_{C12}}{2}\sqrt{e^2{\text{cos}}^2\left({\theta }_{C1}+\frac{t{\theta }_{C12}}{2}\right)+{\text{sin}}^2\left({\theta }_{C1}+\frac{t{\theta }_{C12}}{2}\right)}}{\text{sin}\frac{{\theta }_{C12}}{2}\sqrt{e^2{\text{cos}}^2\left({\theta }_{C1}+\frac{{\theta }_{C12}}{2}\right)+{\text{sin}}^2\left({\theta }_{C1}+\frac{{\theta }_{C12}}{2}\right)}}, \\ {f}_y(t)=\frac{\text{sin}\frac{t{\theta }_{C12}}{2}\sqrt{e^2{\text{sin}}^2\left({\theta }_{C1}+\frac{t{\theta }_{C12}}{2}\right)+{\text{cos}}^2\left({\theta }_{C1}+\frac{t{\theta }_{C12}}{2}\right)}}{\text{sin}\frac{{\theta }_{C12}}{2}\sqrt{e^2{\text{sin}}^2\left({\theta }_{C1}+\frac{{\theta }_{C12}}{2}\right)+{\text{cos}}^2\left({\theta }_{C1}+\frac{{\theta }_{C12}}{2}\right)}}. \end{gathered}$$ (39)

Eq.(39) is the standard form of interpolants that will be used in the example section. The interpolation of the translation vectors become the following linear form when substitute ${\overline{r}}_x={\overline{r}}_y=0$ into (36) and (37):

$$\begin{gathered} C_{mx}\left(t\right)=C_{1x}+f_x\left(t\right)\left(C_{2x}-C_{1x}\right), \\ {C}_{my}\left(t\right)=C_{1y}+f_y\left(t\right)\left(C_{2y}-C_{1y}\right). \end{gathered}$$ (40)

When $e=1$, Eq.(39) reduces to (35) and the path of the above interpolation becomes a straight line.

5. Examples

In the example, we compare the interpolating motions of two objects with the same starting and end positions but with different shape parameters ($e=0.1$ and $e=0.5$). The two given positions are ${\text{T}}_{C1}=\left(2,4,0^{°}\right)$ and ${\text{T}}_{C2}=\left(20,10,-{60}^{°}\right)$, respectively. The resulting interpolating motions are shown in Figure 4 and Figure 5. Figure 6 compares the paths of the interpolating motions as $e$ varies in the range (0, 1). The black straight dash line is the linear interpolation motion and the black circular curve is the pure rotation, which are the two most common ways to describe the motion between two positions in kinematics without incorporating the shape factors. All the other colorful curves are the shape dependent interpolating motions with various values of $e$. As the values of $e$ decrease, the paths of motion increasingly move away from that of the straight-line motion. This shows that the shape of the object rather than its size has effects on the motion. The interpolants used in this example are shown in Figure 7.

FIGURE 4.

An interpolating motion for $e=0.1$.

FIGURE 5.

An interpolating motion for $e=0.5$.

FIGURE 6.

Interpolating motion with various values of $e$.

FIGURE 7.

The interpolants $f_x\left(t\right)$ and $f_y\left(t\right)$ for various values of $e$.

6. Conclusion

In this paper we applied and extended the concept of a shape dependent distance measure between two planar displacements.

The resulting distance measure combines translation and rotation parameters in terms of the area moments of inertia of the object. After decomposing the shape dependent distance in two orthogonal directions, we obtained shape dependent motion interpolants in these directions. The resulting interpolants have similarities to the well-known spherical linear interpolants widely used in computer graphics. The shape parameters can be used to adjust the path of the interpolating motion.

Appendices

APPENDIX A. Derivation of Distance and Components

By subtracting (23) from (24), we obtain the translation components $\left(\Delta x,\Delta y\right)$ in $X$ and $Y$ directions. Substituting them into the integration shown below

$$D_{12x}^2=\frac{1}{S}{\iint }_S\Delta x^{2}dS,D_{12y}^2=\frac{1}{S}{\iint }_S\Delta y^{2}dS$$ (41)

and after some algebra, we obtain

$$\begin{gathered} D_{12x}^2=\frac{1}{S}{{\iint }_S\left[\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)r_x-\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)r_y+d_{2x}-d_{1x}\right]}^{2}dS, \\ =\frac{1}{S}{\iint }_S\left[{\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{r}_x^2+{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{r}_y^2-2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)r_x{r}_y+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2x}-d_{1x}\right)r_x-2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2x}-d_{1x}\right)r_y+{\left(d_{2x}-d_{1x}\right)}^2\right]dS, \end{gathered}$$ (42)

$$D_{12y}^2=\frac{1}{S}{{\iint }_S\left[\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)r_x-\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)r_y+d_{2y}-d_{1y}\right]}^{2}dS,=\frac{1}{S}{\iint }_S\left[{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{r}_x^2+{\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{r}_y^2+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)r_x{r}_y+2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2y}-d_{1y}\right)r_x+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2y}-d_{1y}\right)r_y+{\left(d_{2y}-d_{1y}\right)}^2\right]dS.$$ (43)

As the derivations for $D_{12x}^2$ and $D_{12y}^2$ are basically the same. In what follows, we only show the derivation for $D_{12x}^2$ to illustrate the procedure. The first three terms in Eq. (42) are related to the area moments of inertia and are reorganized as

$$\begin{gathered} {\iint }_S\left[{\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{r}_x^2\right]dS+{\iint }_S\left[{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{r}_y^2\right]dS-{\iint }_S\left[2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)r_x{r}_y\right]dS \\ ={\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{\iint }_S{r}_x^{2}dS+{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{\iint }_S{r}_y^{2}dS \\ ={\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{I}_y+{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{I}_x-2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)I_{xy}, \end{gathered}$$ (44)

where $I_x,I_y,I_{xy}$ are the area moments of inertia in the moving coordinate frame.

The forth and fifth terms in Eq. (42) are related to the centroid of the object with respect to the moving frame. They can be transformed into

$$\begin{gathered} {\iint }_S\left[2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2x}-d_{1x}\right)r_x\right]dS-{\iint }_S\left[2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2x}-d_{1x}\right)r_y\right]dS=2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\iint }_S{r}_{x}dS-2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\iint }_S{r}_{x}dS \\ =2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\overline{r}}_{x}S-2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\overline{r}}_{y}S, \end{gathered}$$ (45)

where $\left({\overline{r}}_x,{\overline{r}}_y\right)$ represents the position of the centroid of the object, which is also the origin of PCF, relative to the moving coordinate frame.

The last term is simply the translation in the X direction:

$${\iint }_S{\left(d_{2x}-d_{1x}\right)}^{2}dS={\left(d_{2x}-d_{1x}\right)}^{2}S.$$ (46)

Substitute (44) (45) (46) into (42) and apply the same procedure to $D_{12y}^2$ to obtain the following expressions for the two components of shape dependent distance:

$$D_{12x}^2={\left(d_{2x}-d_{1x}\right)}^2+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\overline{r}}_x-2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2x}-d_{1x}\right){\overline{r}}_y+\frac{1}{S}\left[{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{I}_x+{\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{I}_y-2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)I_{xy}\right],$$ (47)

$$D_{12y}^2={\left(d_{2y}-d_{1y}\right)}^2+2\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\left(d_{2y}-d_{1y}\right){\overline{r}}_x+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(d_{2y}-d_{1y}\right){\overline{r}}_y+\frac{1}{S}\left[{\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)}^2{I}_x+{\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)}^2{I}_y+2\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)I_{xy}\right].$$ (48)

The moments of inertia as well as the products of inertia can be expressed in terms of their principal components by the following coordinate transformation:

$$\begin{gathered} I_x={\text{cos}}^2\varphi I_u+{\text{sin}}^2\varphi I_v+2\text{sin}\varphi \text{cos}\varphi I_{uv}+S{\overline{r}}_y^2, \\ {I}_y={\text{sin}}^2\varphi I_u+{\text{cos}}^2\varphi I_v-2\text{sin}\varphi \text{cos}\varphi I_{uv}+S{\overline{r}}_u^2, \\ {I}_{xy}=-\text{sin}\varphi \text{cos}\varphi I_u+\text{sin}\varphi \text{cos}\varphi I_v+\left({\text{cos}}^2\varphi -{\text{sin}}^2\varphi \right)I_{uv}+S{\overline{r}}_x{\overline{r}}_y. \end{gathered}$$ (49)

where $I_u,I_v,I_{uv}$ are the area moment of inertia in PCF. Since the product of inertia $I_{uv}=0$ relative to PCF, the final transformation for moment of inertia is

$$\begin{gathered} I_x={\text{cos}}^2\varphi I_u+{\text{sin}}^2\varphi I_v+S{\overline{r}}_y^2, \\ {I}_y={\text{sin}}^2\varphi I_u+{\text{cos}}^2\varphi I_v+S{\overline{r}}_x^2, \\ {I}_{xy}=-\text{sin}\varphi \text{cos}\varphi I_u+\text{sin}\varphi \text{cos}\varphi I_v+S{\overline{r}}_x{\overline{r}}_y. \end{gathered}$$ (50)

Substituting (50) into (47) (48), and after some algebra, the components of shape dependent distance in $X$ and $Y$ can be obtained as

$$D_{12x}^2={\left[\left(d_{2x}-d_{1x}\right)-2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]}^2+4{\text{sin}}^2\frac{{\theta }_{12}}{2}\left[{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)k_u^2+{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)k_v^2\right],$$ (51)

$$D_{12y}^2={\left[\left(d_{2y}-d_{1y}\right)+2\text{sin}\frac{{\theta }_{12}}{2}\left(\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]}^2+4{\text{sin}}^2\frac{{\theta }_{12}}{2}\left[{\text{sin}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)k_u^2+{\text{cos}}^2\left(\frac{{\theta }_1+{\theta }_2}{2}+\varphi \right)k_v^2\right].$$ (52)

In the above, the radius of gyration, $k_u$ and $k_v$, are used to replace $I_u$ and $I_v$ with the following

$$I_u=k_u^{2}S,I_v=k_v^{2}S.$$ (53)

Adding the two expressions given by (51) and (52), we finally obtain

$$D_{12}^2={\left(d_{2x}-d_{1x}\right)}^2+{\left(d_{2y}-d_{1y}\right)}^2+4{\text{sin}}^2\frac{{\theta }_{12}}{2}\left(k_u^2+k_v^2\right)+4\text{sin}\frac{{\theta }_{12}}{2}\left[\left(d_{2y}-d_{1y}\right)\left(\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x-\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right.\left.-\left(d_{2x}-d_{1x}\right)\left(\text{sin}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_x+\text{cos}\frac{{\theta }_1+{\theta }_2}{2}{\overline{r}}_y\right)\right]+4{\text{sin}}^2\frac{{\theta }_{12}}{2}\left({\overline{r}}_x^2+{\overline{r}}_y^2\right).$$ (54)

Furthermore, substitute (9) into (54) and (51)(52), the final expressions for shape dependent distance and its components are shown in (25) and (27)(28).

APPENDIX B. Transformation between PCF and Moving Frame

Let ${\text{T}}_{C1}=\left(C_{1x},C_{1y},{\theta }_{C1}\right)$ and ${\text{T}}_{C2}=\left(C_{2x},C_{2y},{\theta }_{C2}\right)$ be the displacements of PCF with respect to the fixed coordinate frame at two positions. ${\text{T}}_{Cm}=\left(C_{mx},C_{my},{\theta }_{Cm}\right)$ is the interpolated position varies with $t$. Transformations between PCF and the moving frame are as follows

$${\text{C}}_1={\text{d}}_1+\left[A_1\right]\overline{\text{r}},{\text{C}}_2={\text{d}}_2+\left[A_2\right]\overline{\text{r}},$$ (55)

$${\text{C}}_{\text{m}}\left(t\right)={\text{d}}_{\text{m}}\left(t\right)+\left[A_m\left(t\right)\right]\overline{\text{r}}.$$ (56)

Eq.(56) can be separated into

$$\begin{gathered} C_{mx}\left(t\right)=d_{mx\left(t\right)}+{\overline{r}}_x\text{cos}{\theta }_m\left(t\right)-{\overline{r}}_y\text{sin}{\theta }_m\left(t\right), \\ {C}_{my}\left(t\right)=d_{my\left(t\right)}+{\overline{r}}_x\text{sin}{\theta }_m\left(t\right)+{\overline{r}}_y\text{cos}{\theta }_m\left(t\right). \end{gathered}$$ (57)

Substitute (36)(37) into the above equations and simplify to obtain

$$C_{mx}\left(t\right)=\left(1-f_x\left(t\right)\right)d_{1x}+f_x\left(t\right)d_{2x}+{\overline{r}}_x\left[\text{cos}{\theta }_1+f_x\left(t\right)\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\right]-{\overline{r}}_y\left[\text{sin}{\theta }_1+f_x\left(t\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\right],$$ (58)

$$C_{my}\left(t\right)=\left(1-f_y\left(t\right)\right)d_{1y}+f_y\left(t\right)d_{2y}+{\overline{r}}_x\left[\text{sin}{\theta }_1+f_y\left(t\right)\left(\text{sin}{\theta }_2-\text{sin}{\theta }_1\right)\right]+{\overline{r}}_y\left[\text{cos}{\theta }_1+f_y\left(t\right)\left(\text{cos}{\theta }_2-\text{cos}{\theta }_1\right)\right].$$ (59)

From (55), we can obtain $d_{1x},d_{1y},d_{2x},d_{2y}$ in terms of $C_{1x},C_{1y},C_{2x},C_{2y}$, which can be substituted into (58)(59). The final result for interpolated position is shown in (40).