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Last time

- What did we talk about last time?
- Texturing in XNA
- Homogeneous notation
- Geometric techniques
- Lines (2D and 3D)
- Planes

Implicit planes

- The implicit equation for a plane is just like the implicit for a line, except with an extra dimension
- Let n be a vector (a, b, c)
- Let p be a point (x, y, z)
- p is on plane π if and only if
- n • p + d = 0
- Since n is the normal of the plane we can compute it in a couple of ways:
- n = s x t where s and t span the plane
- n = (u – w) x (v – w) where u, v, and w are noncollinear points in the plane

Implicit plane function

- Again, it works just like a line
- If we write f(p) = n • p + d
- Assuming q plane π
- f(p) = 0 iffp π
- f(p) > 0 iffp lies on the same side as the point q + n
- f(p) < 0 iffp lies on the same side as the point q - n
- f(p) again gives a (scaled) measure of the perpendicular distance from p to π
- Distance(p) = f(p)/||n||
- This implies that Distance(0) = d, making d the shortest distance from the origin to the plane

Cross product

- The cross product of two vectors finds a vector that is orthogonal to both
- For 3D vectors u and v in an orthonormal basis, the cross product w is:

Practice with a plane

- Plane defined by:
- (1, 3, 2)
- (2, 5, 2)
- (3, 8, 2)
- Where is (3, 3, 2)?
- What about (3, 4, 5)?

Transforms

- A transform is an operation that changes points, vectors or colors
- We can use them to position and animate objects, lights, and cameras
- A linear transform is one that holds over vector addition and scalar multiplication
- Rotation
- Scaling
- Can be represented by a 3 x 3 matrix

Affine transforms

- Adding a vector after a linear transform makes an affine transform
- Affine transforms can be stored in a 4 x 4 matrix using homogeneous notation
- Affine transforms:
- Translation
- Rotation
- Scaling
- Reflection
- Shearing

Translation

- Move a point from one place to another by vector t = (tx, ty, tz)
- We can represent this with translation matrix T

Rotation

- Rotation, like translation, is a rigid-body transform (points don\'t change distance from each other and handedness doesn\'t change)
- An orientation matrix is used to define up and forward (usually for a camera)
- We usually express rotation in terms of 3 separate x, y, and z rotation matrices

Rotation around a point

- Usually all the rotations are multiplied together before translations
- But if you want to rotate around a point
- Translate so that that point lies at the origin
- Perform rotations
- Translate back

Scaling

- Scaling is easy and can be done for all axes as the same time with matrix S
- If sx = sy = sz, the scaling is called uniform or isotropic and nonuniform or anisotropic otherwise

Shearing

- A shearing transform distorts one dimension in terms of another with parameter s
- Thus, there are six shearing matrices Hxy(s), Hxz(s), Hyx(s), Hyz(s), Hzx(s), and Hzy(s)
- Here\'s an example of Hxz(s):

To make a shearing matrix

- To make Hij(s), start with the identity matrix and put s in row i, column j
- See how the only the affected dimension changes

Concatenation of transforms

- Because matrix multiplications are not commutative, order of transforms matters
- Still, parts (or the entirety) of the transform can be precomputed and stored as a single matrix
- Note that matrices are applied from right to left, thus TRSp scales point p, rotates it, then translates it

Non-commutativity of transforms

- This example from the book shows how the same sets of transforms, applied in different orders, can have different outcomes

Rigid-body transforms

- A rigid-body transform preserves lengths, angles, and handedness
- We can write any rigid-body transform X as a translation matrix T(t) multiplied by a rotation matrix R

Inverse of a rigid-body transform

- Because R is orthogonal, its inverse is its transpose
- Because of the nature of a translation, its inverse is just its negative
- Thus, the inverse of X is
- X-1 = (T(t)R)-1 = R-1T(t)-1 = RTT(-t)

Next time…

- Normal transforms
- Inverses
- Euler transform
- Matrix decomposition
- Rotation around an arbitrary axis
- Quaternions

Reminders

- Keep reading Chapter 4
- Finish Project 1, due tonight by 11:59

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