CS450: Computer Graphics

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CS450 Computer Graphics
Geometric 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.
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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
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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
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Dot and Cross Products
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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
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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
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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 lt
0, T lies beneath the plane
0, T lies on the plane
gt 0, T lies above the
plane
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Planes
A vector, n, which is orthogonal to both u and v
can be computed as
n u x v This vector is called the
surface normal.