![Page 1: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/1.jpg)
Vectors Tools for Graphics
![Page 2: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/2.jpg)
To review vector arithmetic, and to relate vectors to objects of interest in graphics.
To relate geometric concepts to their algebraic representations.
To describe lines and planes parametrically.To distinguish points and vectors properly.To exploit the dot product in graphics topics.To develop tools for working with objects in 3D
space, including the cross product of two vectors.
Vector, Geometry and CG
![Page 3: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/3.jpg)
Computer graphics objects
Objects to be drawn Shape position orientation
fundamental mathematical discipline to aid graphics is vector analysis transformation
![Page 4: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/4.jpg)
Why vector analysis
![Page 5: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/5.jpg)
2-D and 3-D coordinate systems
![Page 6: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/6.jpg)
Vector Review
The difference between two points is a vector: v = Q - P;
![Page 7: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/7.jpg)
Vector and Point
Turning this around, we also say that a point Q is formed by displacing point P by vector v; we say that v offsets P to form Q. Algebraically, Q is then the sum:
Q = P + v.
The sum of a point and a vector is a point: P + v = Q.
![Page 8: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/8.jpg)
Vector representation
At this point we represent a vector through a list of its components: an n-dimensional vector is given by an n-tuple:
w = (w 1 , w 2 , . . . , w n )
![Page 9: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/9.jpg)
Operation with Vectors
AddScale
![Page 10: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/10.jpg)
Linear Combination of Vectors
a v + b w
![Page 11: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/11.jpg)
Affine combination of vectors
A linear combination of vector is affine combination if
ex: 3 a + 2 b - 4 c
![Page 12: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/12.jpg)
Convex combination of Vectors
Plus one more requirementai >= 0 I = 1…m.3a+.7b1.8a -.8bThe set of coefficients a 1 , a 2 , . . . , a m
is sometimes said to form a partition of unity, suggesting that a unit amount of material is partitioned into pieces.
![Page 13: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/13.jpg)
The Magnitude of a vector
Note that if w is the vector from point A to point B, then |w| will be the distance from A to B
![Page 14: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/14.jpg)
Unit vector
It is often useful to scale a vector so that the result has a length equal to one. This is called normalizing a vector, and the result is known as a unit vector. For example, we form the normalized version of a, denoted , by scaling it with the value 1/|a|:
Ex. a = (3, -4),
![Page 15: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/15.jpg)
The dot product
![Page 16: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/16.jpg)
![Page 17: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/17.jpg)
![Page 18: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/18.jpg)
The Angle Between Two Vectors.
![Page 19: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/19.jpg)
The Sign of b.c and Perpendicularity.
![Page 20: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/20.jpg)
![Page 21: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/21.jpg)
The 2D Perp Vector.
![Page 22: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/22.jpg)
The perp dot product
![Page 23: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/23.jpg)
Orthogonal Projections
![Page 24: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/24.jpg)
Calculate K and M
![Page 25: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/25.jpg)
The distance from C to The Line
the distance from a point C to the line through A in the direction v is:
![Page 26: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/26.jpg)
The Cross Product of Two Vectors
The cross product (also called the vector product) of two vectors is another vector. It has many useful properties, but the one we use most often is that it is perpendicular to both of the given vectors. The cross product is defined only for three-dimensional vectors.
![Page 27: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/27.jpg)
Properties
![Page 28: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/28.jpg)
Normal
![Page 29: Vectors Tools for Graphics. To review vector arithmetic, and to relate vectors to objects of interest in graphics. To relate geometric concepts to](https://reader036.vdocuments.site/reader036/viewer/2022062314/56649e175503460f94b03099/html5/thumbnails/29.jpg)
Finding the Normal to a Plane