Computer Graphics Geometric Objects and Transformations
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Transcript of Computer Graphics Geometric Objects and Transformations
![Page 1: Computer Graphics Geometric Objects and Transformations](https://reader036.fdocuments.net/reader036/viewer/2022082413/56812c71550346895d910982/html5/thumbnails/1.jpg)
Computer GraphicsGeometric Objects and Transformations
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Scalars, Points, and Vectors:
A scalar is a magnitude only. Ex: -3.5
A vector is a magnitude and a direction. Vectors are represented pictorially as a DIRECTED LINE SEGMENT. The length of the vector represents its magnitude.
Identical vectors
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Scalars, Points, and Vectors:
Two vectors are ADDED using the HEAD-TO-TAIL rule, and addition is commutative.
A vector is NOT ANCHORED in space. A POINT is anchored in space as the HEAD of a vector extending outward from the ORIGIN of the space.
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Lines:
if P and Q are points in an affine space, the set of points of the form
R(t) =(1- t) P + t Q
form a line passing through P and Q.
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Lines:
The parametric form of a line in space is given by:
P( ) = Q + v
Q is the anchor point
v is a vector that points in the direction of the line.
is a scalar that varies
P( ), a function of , is the set of all points along the line passing through Q and having the direction v.
( Note that this notation is slightly different than the text.)
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Convexity:
An object is CONVEX if any point on a line segment between any two points in the object is also in the object.
The CONVEX HULL of an object is the smallest convex object which contains the original.
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Dot and Cross Products:
The square of the magnitude of a vector v is given by
the dot product:
| v | = v v.2
v v = v v + v v + … + v v2 21 n. 1 n
v =v1
v2
vn
...
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Dot and Cross Products:
The cosine of the angle between two vectors u and v is given by
cos 0 = |u | |v|
u v = u v + u v + … + u v2 21 n. 1 nv =
v1
v2
vn
...
.u v
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Dot and Cross Products:
The angle between two vectors u and v can also be computed using the magnitude of the cross product
|sin 0| = |u | |v| |u x v|
v =v1
v2
v3
u =u1
u2
u3
u x v = u v - u v 3 22 3
u v - u v 1 33 1
u v - u v 2 11 2
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Planes:
if P, Q and R are three points in an affine space, and they are not coliniear, then the plane defined by P, R and Q is:
F(s,t) = (1-s)((1- t) P + t Q)+ s R
A plane can also be described in terms of a point, P, and two non-parallel vectors, u and v
A = P + u + v
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Planes:
The plane equation can also be given as follows:
(a,b,c) (x-x0,y-y0,z-z0) = 0
where n = (a,b,c) plane normal
T = (x,y,z) T represents any test point
P = (x0,y0,z0) a known point in the plane
.
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Planes:
ax + by + cz + d = 0 ,where d = -(ax0 + by0 + cz0)
If we evaluate the left side for a given point T = (x,y,z) in 3D space and the result is
< 0, T lies beneath the plane
= 0, T lies on the plane
> 0, T lies above the plane
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Planes:
A plane can also be described in terms of a point, P, and two non-parallel vectors, u and v
A = P + u + v
A vector, n, which is orthogonal to both u and v can be computed as
n = u x v n