1. VIRTUAL ENVIRONMENT

2. VIRTUAL ENVIRONMENT The Dynamics of Numbers The Animation of Objects
Numerical Interpolation
Linear Interpolation
Non-Linear Interpolation
The Animation of Objects
Linear Translation
Non-Linear Translation
Shapes and Objects in between

3. The Dynamics of Numbers
Virtual Environment – A Complex Numerical DB
It changes one number to another
Important: Numerical Envelope of the change
Example: In animating the bouncing of a ball, it’s centroid is used to guide its motion
Realistic motions are simulated by numerical interpolation
Numerical Interpolation – Linear and Non-Linear

4. Numerical Interpolation
Interpolation produces a function that matches the given data exactly.
The function then can be utilized to approximate the data values at intermediate points
It is also used to produce a function for which values are known only at discrete points, either from measurements or calculations.

5. Numerical Interpolation
Given data points
Obtain a function, P(x)
P(x) goes through the data points
Use P(x)
To estimate values at intermediate points
For example, given data points:
At x0 = 2, y0 = 3 and at x1 = 5, y1 = 8
Find the following:
At x = 4, y = ?

6. Numerical Interpolation

7. Linear Interpolation
Linear Interpolation - Simplest interpolation method
Apply LI(a sequence of points)  A polygonal line where each straight line segment connects two consecutive points of the sequence
Therefore, for every segment (P,Q)
P(x) = (1 - x)P + xQ where x ϵ [0,1]
By varying x from 0 to 1, we get all the intermediate points between P and Q
P(x) = P for x = 0 and P(x) = Q for x = 1.
We get points on the line defined by P, Q, when 0 ≤ x ≤ 1

8. Non-Linear Interpolation
A linear interpolation ensures that equal steps in the parameter x give rise to equal steps in the interpolated values
It is often required that equal steps in x give rise to unequal steps in the interpolated values
We can achieve this using a variety of mathematical techniques
For example, we could use trigonometric functions or polynomials to achieve this

9. Trigonometric interpolation
sin2(x) + cos2(x) = 1
If x varies between 0 and π/2
cos2(x) varies between 1 and 0, sin2(x) varies between 0 and 1
which can be used to modify the two interpolated values n1 and n2 as follows
n=n1cos2(t)+n2sin2(t) for 0 ≤ t ≤ π/2

10. Trigonometric interpolation

11. Trigonometric interpolation
Let n1 = 1 and n2 = 3

12. Cubic interpolation

13. Cubic interpolation

14. Cubic interpolation

15. The Animation of Objects
Newton’s Laws of Motion provide a useful framework to predict an object’s behavior under dynamic conditions
This scenario can be simulated within a Virtual Environment
It can be achieved through a simple linear translation of objects

16. Uses of Translation Modeling transformations Viewing transformations
build complex models by positioning simple components
transform from object coordinates to world coordinates
Viewing transformations
placing the virtual camera in the world
i.e. specifying transformation from world coordinates to camera coordinates
Animation
vary transformations over time to create motion

17. Linear Translation
Consider an object located at VE’s origin and assigned a speed S0 across the XY plane
To simulate its sliding movement, the x & z coordinates of the object must be modified (tnow – tprev)V0
The object’s new velocity can be computed after it is bouncing off the boundary.

18. Linear Translation

19. Non-Linear Translation
Consider an object moving along the x-axis in 1s, pause momentarily and then returns to its original position in 2s
The non-linear movement of the object can be simulated by computing the x-translation as a function of time
At time t1, the translation begins. At time t2, i.e.,(t1+1) it pauses momentarily and at time t3, i.e., (t1+3), it comes to rest
t = T – t1 while t1 ≤ T ≤ t T – Current time
t = (T – t1 – 1)/2 while t2 ≤ T ≤ t3 t- Control parameter

20. Euler Angles
Orientation of an object in computer graphics is often described using Euler Angles
RxRyRz
Any axis order will work and could be used.
Yaw, Pitch & Roll

21. Problem with Euler Angles
Rotations not uniquely defined
Ex: (z, x, y) [roll, yaw, pitch] = (90, 45, 45) = (45, 0, -45) takes positive x-axis to (1, 1, 1)
Cartesian coordinates are independent of one another, but Euler angles are not
Remember, the axes stay in the same place during rotations
Gimbal Lock
Term derived from mechanical problem that arises in gimbal mechanism that supports a compass or a gyro
Second and third rotations have effect of transforming earlier rotations, we lose a degree of freedom
ex: Rot x, Rot y, Rot z
If Rot y = 90 degrees, Rot z is equivalent to -Rot x
With Euler Angles, knowing the transformations for the entire rotation does not help us with the intermediate rotations. Each is a unique set of three rotations about the x,y and z axes

22. Quaternions Invented by Sir William Hamilton (1843)
Do not suffer from Gimbal Lock
Provide a natural way to interpolate intermediate steps in a rotation about an arbitrary axis
Are used in many position tracking systems and VR software support systems

23. Quaternions
You can think of quaternions as an extension of complex numbers where there are three different square roots of -1.
q = w + i x + j y + k z where
i = (-1), j = (-1), k = (-1)
i*j = k, j*k = i, k*i = j
j*i = -k, k*j = -i, i*k = -j
You could also think of q as a value in four-dimensional space, q = [w, x, y, z]
Sometimes written as q = [w, v] where w is a scalar and v is a vector in 3-space

24. Shape and Object Inbetweening
Shape Inbetweening
It is a technique in the cartoon industry to speed up the process of creating art works
The ‘inbetween’ objects are derived from two key images drawn by a skilled animator
The images are called as Key Frames
The key frames shows an object in two different positions
The number and position of inbetweenig images determines the dynamics of final animation

25. Shape Inbetweening

26. Object Inbetweening
By inbetweening the z-coordinate, the above technique can be applied to 3D contours
We cannot expect to interpolate between two different complex objects and geometrically consistent
Given suitable geometric definitions, it is possible to transform one object into another and create subtle animation