Objective

2

Unit quaternion represents rotation more compactly and intuitively than rotation matrix and is superior to Euler angle parameterization in avoiding gimbal lock. This study constructs a class of

$${C^3}$$

-continuous unit quaternion interpolation spline curves. The curves in unit quaternion space are an extension of a class of quintic polynomial interpolation spline curves in Euclidean space. They maintain high continuity and interpolating properties and are suitable for controlling the orientation of a rigid body in keyframe animation.

Method

2

Rigid keyframe animation usually has two or more keyframes (nodes) to be connected

and the rigid body needs to pass through the keyframes accurately. When the second derivatives of the orientation curve are discontinuous at some nodes in the spline curve

the angular acceleration of the object will suffer from a large jump

thereby resulting in undesirable rotation effects that are inconsistent with the expectation. Therefore

how to construct a high-order continuous unit quaternion curve is important in computer animation. This study first gives the definition and properties of quaternions and then selects proper quintic polynomial-blending functions to achieve a smooth interpolating orientation curve. Interpolation spline curves are constructed in

$${\mathit{\boldsymbol{R}}^3}$$

space by taking such quintic polynomial-blending functions as the base functions. The constructed curves can pass through the given data points and are

$${C^3}$$

-continuous when the knot vector is evenly distributed. We extend the spline curve in

$${\mathit{\boldsymbol{R}}^3}$$

space into

$${S^3}$$

space. Therefore

the unit quaternion interpolation spline curve maintains many similar important properties to those of the quintic polynomial interpolation spline curve in Euclidean space

such as interpolation and high-order continuity. Inspired by Kim et al.

[5]

we take the cumulative forms of the constructed blending functions of the interpolation spline curves in Euclidean space as the base functions to construct unit quaternion interpolation spline curves in

$${S^3}$$

space. Specifically

the unit quaternion interpolation spline curve is the product of several exponential functions

in which the exponents are cumulative forms of the blending functions

and the bases are constant unit quaternions that are the given keyframe orientations and the angular velocities between every two adjacent given keyframe orientations. Using this ability

the produced unit quaternion curves behave similarly to the spline curves in Euclidean space. For example

they not only pass through a given sequence of orientations (denoted by unit quaternions) exactly without the iterative process in solving the quaternionic nonlinear system of equations

as used in the classical B-spline unit quaternion interpolatory curves to obtain the control points from given data points but also can possess

$${C^3}$$

continuity. A strict proof of these properties is given. An application example is provided to show that the constructed unit quaternion curve can play a role in the keyframe animation design.

Result

2

We provide a 3D keyframe interpolation animation instance and comparison experiments to illustrate the feasibility and effectiveness of the proposed unit quaternion orientation curve construction method. From the application example

we observe that the proposed curve automatically passes through a given sequence of keyframe orientation accurately. The postures of cherries are controlled by the unit quaternion curve. Red cherries are randomly given keyframes. The orientations of intermediate green cherries are determined by three different types of unit quaternion spline curves. The position curve of the cherries adopts the same cubic uniform B-spline curves. Unlike Nielson's iterative construction scheme of unit quaternion uniform B-spline interpolation curves

the proposed scheme avoids solving the nonlinear system of equations over quaternions to obtain the control points

which greatly improves computational efficiency. Unlike Su et al.'s algebraic trigonometric blending interpolation quaternion spline curves

our scheme only uses polynomial bases

which run faster than trigonometric functions. The time required for the animation example by using our scheme decreases by approximately 73% and 33% compared with those by Nielson's and Su et al.'s method

respectively. Moreover

the continuity of the curve produced by the proposed scheme is

$${C^3}$$

higher than the

$${C^2}$$

-continuous curves constructed by the other two methods

which means that the changes in the orientation of the rigid body in the animation will be more natural.

Conclusion

2

Simulation results demonstrate that the proposed method is effective for rigid-body keyframe animation design and particularly applicable for design occasions with high real-time and fluency demands. Although the proposed method has the advantages of automatically accurate interpolation

high continuity

and fast running

this paper does not discuss the fairness of the curve. From numerous experimental results

we deduce that the curve fairness produced by the proposed method is not as good as the fairness of unit quaternion uniform B-spline curves. The next step of our work will be to study the construction method of a quaternion interpolating spline curve from the aspect of energy optimization

such as minimizing torque energy or curvature.