## Computer animation refers to the computer-based generation of animations .

It uses the means of computer graphics and supplements them with additional nxt digital techniques.

Sometimes a distinction is made between computer-based and computer-generated animation. between which the animation software interpolates in order to maintain smooth movements, the latter works with a three-dimensional scene from which images are directly generated using the means of image synthesis .Keyframes and interpolation

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Scene from the computer-generated short film Big Buck Bunny (2008)

The technique of keyframe animation ( keyframing ) originally comes from the production of animated films . Not all individual pictures are drawn by hand, but only so-called key pictures, which roughly specify the sequence of movements. The individual frames between the key frames are automatically calculated by interpolation techniques, this is also called tweening .

In the case of computer-generated animations, the term “key frame” is misleading in that the interpolation is not based on the complete frames. Instead, various parameters of the scene are set, such as the positions of the object centers, their colors and scaling, the camera position and direction of view or the intensity of the light sources. Different key frames can also be selected for different parameters.

### Interpolation and nxt digital

**Different TCB splines**

In order to avoid sudden changes in speed, the interpolation curve of the key values is usually chosen so that its derivation is continuous . A higher continuous differentiability is usually not necessary, since the second derivative (acceleration) often changes abruptly in nature. These properties make Catmull-Rom splines a good choice for animation curves. Many animation systems allow the animator to fine-tune the animation curve by specifically adapting the tangents to the key values. This is often done by controlling tension, continuity and bias (TCB). In addition, the Catmull-Rom to Kochanek-Bartels splines with which these three parameters can be set for each key value.

If a movement path has been obtained by digitization, it must first be smoothed out before it can be used in order to remove jumps and noise.

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### Curve parameterization

The same parametric distances, marked here by white arrows, do not match the same curve distances (black arrows)

Simply describing the animation curve is generally not sufficient if not only the position but also the speed of an object along a path is to be checked. This is because by fitting a curve to multiple points, nothing is said about the speed between those points. The parameterization of the curve types usually used for animation does not match the actual distance along a curve.

One way to control the speed of an object is to have the animator determine an additional distance-time function that specifies how far an object should have moved along the curve at a certain point in time.

Speed-time or even acceleration-time functions are also possible. In any case, the animation program must internally convert the curve length along an animation curve into its parameter representation. There is no analytical formula for most spline types, so approximation methods such as the Newton method must be used.

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The most common speed profile in animation is ease-in / ease-out. An object accelerates from the starting point, reaches a maximum speed, and finally brakes again to the end point. This behavior can be modeled by a segment of the sine function .

**Rotations**

In addition to displacement, rotation is the only transformation that maintains the shape of an object; it therefore plays a major role in the animation of rigid bodies. Special methods are used to interpolate three-dimensional rotations.

A simple way of specifying the rotation of an object is by means of Euler angles . When these angles are animated, the gimbal lock problem can occur. This effect occurs when one of the three axes of rotation coincides with another, thereby losing one degree of freedom .

To avoid this problem, quaternions are used in the computer animation to formulate rotations. Quaternions form a four-dimensional space for which operations such as addition and multiplication are defined. In order to rotate a point, it is first represented as a quaternion, the rotation is used in quaternion space, and then converted back into the usual Cartesian coordinates . Successive rotations in the quaternion space correspond to products of quaternions.

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Rotations are usually expressed by unit quaternions, which can be thought of as points on a four-dimensional unit sphere . The interpolation on the four-dimensional unit sphere is also Slerpcalled. Since it is mathematically nxt digital very complex, interpolation between quaternions is often only linear. The intermediate steps are then normalized in order to project them back onto the four-dimensional unit sphere. More consistent results can be achieved by using the De-Casteljau It uses the means of computer graphics and supplements them with additional nxt digital techniques. algorithm so that multiple linear interpolations are performed.

If an object is to follow a movement path, nxt digital it is often expected that it will not only be moved, but also that its orientation will follow the path. The relationship between the orientation of an object and the curve properties can be expressed by the frenetic formulas .

There are some special cases to be considered in which the formulas cannot be used, for example in curve sections without curvature. If the camera is to be moved, a center of interest is often given, which should always be in the center of the image.

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