1. Mathematics for Computer Graphics
Chun-Yuan Lin CG
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2. Coordinate Reference Frames
See the powerpoint: Coordinate Reference Frames.ppt CG
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3. Points and Vectors (1)
There is a fundamental difference between the concept of a geometric point and that of a vector.
A point is a position specified with coordinate values in some reference frame. (depend on the choice for the frame of refernece)
A vector has properties that are independent of any particular coordinate system.
Point Properties P y
Frame B x
Frame A CG
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4. Points and Vectors (2) Vector Properties
We can define a vector as the difference between two point positions.
Vx and Vy are the projection V onto the x and the y axes.
We can obtain these same vector components using two other point positions in the same coordinate reference frames.
A vector has no fixed position within a coordinate system.
We can describe a vector as a directed line segment that has two fundamental properties: magnitude and direction. P2 V P1 CG
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5. Points and Vectors (3) Magnitude:
We can specify the vector direction in various ways, such as
A vector has the same magnitude and direction within a single coordinate system.
If we transform the vector to another reference frame, the value for its components and direction within that reference frame may change.
For a three-dimensional Cartesian vector representation CG
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6. Points and Vectors (4)
We can give the vector direction in terms of the direction angles, α, β, γ.
The values cosα, cos β, cos γ are called the direction cosines of the vector.
Vectors are used to represent any quantities that have the properties of magnitude and direction. (force and velocity) z V γ β y α x CG
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7. Points and Vectors (5) Vector Addition and Scalar Multiplication V2
V1+V2 V2 V1 V1 CG
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8. Points and Vectors (6) Scalar Product of two Vectors
This multiplication scheme is called the scalar product or dot product. (inner product)
is the projection of vector V2 in the direction of V1.
In addition to the coordinate-independent form of the scalar product. V2 V1 θ CG
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9. Points and Vectors (7)
The scalar product of two vectors is zero if and only if the two vectors are perpendicular (orthogonal) CG
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10. Points and Vectors (8) Vector Product of Two Vectors Cross product
V1 × V2 V2 V1 u
Cross product CG
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11. Points and Vectors (9) CG
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12. Matrices (1)
A matrix is a rectangular array of quantities, called the elements of the matrix.
We identify matrices according to the number of rows and number of columns. When the number of rows is the same as the number of columns, this matrix is called a square matrix.
An r by c matrix
Row vector
Column vector CG
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13. Matrices (2)
The matrix representation for a three-dimensional vector in Cartesian coordinates as
We use this standard matrix representation for both points and vectors. CG
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14. Scalar Multiplication and Matrix AdditionCG
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15. Matrix Multiplication(1)
The product of two matrices is defined as a generalization of the vector dot product. CG
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16. Matrix Multiplication(2)
AB≠BA
A(B+C)=AB+AC CG
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17. Matrix Transpose
The transpose MT of a matrix is obtained by interchanging rows and columns.
(M1M2)T=M2TM1T CG
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18. Determinant of a Matrix
If we have a square matrix, we can combine the matrix elements to produce a single number called the determinant of the matrix. CG
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19. Matrix Inverse
With square matrices, we can obtain an inverse matrix if and only of the determinant of the matrix is nonzero.
Identity matrix CG
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