chapter 4.1 mathematical concepts. 2 applied trigonometry trigonometric functions defined using...
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Chapter 4.1Mathematical Concepts
2
Applied Trigonometry
Trigonometric functions Defined using right triangle
x
yh
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Applied Trigonometry
Angles measured in radians
Full circle contains 2 radians
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Applied Trigonometry
Sine and cosine used to decompose a point into horizontal and vertical components
r cos
r sin r
x
y
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Applied Trigonometry
Trigonometric identities
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Applied Trigonometry
Inverse trigonometric functions Return angle for which sin, cos, or tan
function produces a particular value If sin = z, then = sin-1 z If cos = z, then = cos-1 z If tan = z, then = tan-1 z
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Applied Trigonometry
Law of sines
Law of cosines
Reduces to Pythagorean theorem when = 90 degrees
b
a
c
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Vectors and Matrices
Scalars represent quantities that can be described fully using one value Mass Time Distance
Vectors describe a magnitude and direction together using multiple values
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Vectors and Matrices
Examples of vectors Difference between two points
Magnitude is the distance between the points Direction points from one point to the other
Velocity of a projectile Magnitude is the speed of the projectile Direction is the direction in which it’s traveling
A force is applied along a direction
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Vectors and Matrices
Vectors can be visualized by an arrow The length represents the magnitude The arrowhead indicates the direction Multiplying a vector by a scalar
changes the arrow’s lengthV
2V
–V
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Vectors and Matrices
Two vectors V and W are added by placing the beginning of W at the end of V
Subtraction reverses the second vector
V
W
V + W
V
W
V
V – W–W
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Vectors and Matrices
An n-dimensional vector V is represented by n components
In three dimensions, the components are named x, y, and z
Individual components are expressed using the name as a subscript:
1 2 3x y zV V V
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Vectors and Matrices
Vectors add and subtract componentwise
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Vectors and Matrices
The magnitude of an n-dimensional vector V is given by
In three dimensions, this is
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Vectors and Matrices
A vector having a magnitude of 1 is called a unit vector
Any vector V can be resized to unit length by dividing it by its magnitude:
This process is called normalization
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Vectors and Matrices
A matrix is a rectangular array of numbers arranged as rows and columns A matrix having n rows and m columns
is an n m matrix At the right, M is a
2 3 matrix
If n = m, the matrix is a square matrix
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Vectors and Matrices
The entry of a matrix M in the i-th row and j-th column is denoted Mij
For example,
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Vectors and Matrices
The transpose of a matrix M is denoted MT and has its rows and columns exchanged:
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Vectors and Matrices
An n-dimensional vector V can be thought of as an n 1 column matrix:
Or a 1 n row matrix:
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Vectors and Matrices
Product of two matrices A and B Number of columns of A must equal
number of rows of B Entries of the product are given by
If A is a n m matrix, and B is an m p matrix, then AB is an n p matrix
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Vectors and Matrices
Example matrix product
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Vectors and Matrices
Matrices are used to transform vectors from one coordinate system to another
In three dimensions, the product of a matrix and a column vector looks like:
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Vectors and Matrices
The n n identity matrix is denoted In For any n n matrix M, the product with
the identity matrix is M itself InM = M MIn = M
The identity matrix is the matrix analog of the number one
In has entries of 1 along the main diagonal and 0 everywhere else
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Vectors and Matrices
An n n matrix M is invertible if there exists another matrix G such that
The inverse of M is denoted M-1
1 0 0
0 1 0
0 0 1
n
MG GM I
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Vectors and Matrices
Not every matrix has an inverse A noninvertible matrix is called
singular Whether a matrix is invertible can
be determined by calculating a scalar quantity called the determinant
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Vectors and Matrices
The determinant of a square matrix M is denoted det M or |M|
A matrix is invertible if its determinant is not zero
For a 2 2 matrix,
deta b a b
ad bcc d c d
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The Dot Product
The dot product is a product between two vectors that produces a scalar The dot product between two
n-dimensional vectors V and W is given by
In three dimensions,
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The Dot Product
The dot product satisfies the formula
is the angle between the two vectors Dot product is always 0 between
perpendicular vectors If V and W are unit vectors, the dot
product is 1 for parallel vectors pointing in the same direction, -1 for opposite
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The Dot Product
The dot product of a vector with itself produces the squared magnitude
Often, the notation V 2 is used as shorthand for V V
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The Dot Product
The dot product can be used to project one vector onto another
V
W
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The Cross Product
The cross product is a product between two vectors the produces a vector The cross product only applies in three
dimensions The cross product is perpendicular to
both vectors being multiplied together The cross product between two parallel
vectors is the zero vector (0, 0, 0)
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The Cross Product
The cross product between V and W is
A helpful tool for remembering this formula is the pseudodeterminant
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The Cross Product
The cross product can also be expressed as the matrix-vector product
The perpendicularity property means
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The Cross Product
The cross product satisfies the trigonometric relationship
This is the area ofthe parallelogramformed byV and W
V
W
||V|| sin
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The Cross Product
The area A of a triangle with vertices P1, P2, and P3 is thus given by
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The Cross Product
Cross products obey the right hand rule If first vector points along right thumb, and
second vector points along right fingers, Then cross product points out of right palm
Reversing order of vectors negates the cross product:
Cross product is anticommutative
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Stop Here
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Transformations
Calculations are often carried out in many different coordinate systems
We must be able to transform information from one coordinate system to another easily
Matrix multiplication allows us to do this
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Transformations
Suppose that the coordinate axes in one coordinate system correspond to the directions R, S, and T in another
Then we transform a vector V to the RST system as follows
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Transformations
We transform back to the original system by inverting the matrix:
Often, the matrix’s inverse is equal to its transpose—such a matrix is called orthogonal
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Transformations
A 3 3 matrix can reorient the coordinate axes in any way, but it leaves the origin fixed
We must at a translation component D to move the origin:
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Transformations
Homogeneous coordinates Four-dimensional space Combines 3 3 matrix and
translation into one 4 4 matrix
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Transformations
V is now a four-dimensional vector The w-coordinate of V determines
whether V is a point or a direction vector If w = 0, then V is a direction vector and
the fourth column of the transformation matrix has no effect
If w 0, then V is a point and the fourth column of the matrix translates the origin
Normally, w = 1 for points
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Transformations
The three-dimensional counterpart of a four-dimensional homogeneous vector V is given by
Scaling a homogeneous vector thus has no effect on its actual 3D value
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Transformations
Transformation matrices are often the result of combining several simple transformations Translations Scales Rotations
Transformations are combined by multiplying their matrices together
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Transformations
Translation matrix
Translates the origin by the vector T
translate
1 0 0
0 1 0
0 0 1
0 0 0 1
x
y
z
T
T
T
M
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Transformations
Scale matrix
Scales coordinate axes by a, b, and c If a = b = c, the scale is uniform
scale
0 0 0
0 0 0
0 0 0
0 0 0 1
a
b
c
M
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Transformations
Rotation matrix
Rotates points about the z-axis through the angle
-rotate
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
z
M
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Transformations
Similar matrices for rotations about x, y
-rotate
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
x
M
-rotate
cos 0 sin 0
0 1 0 0
sin 0 cos 0
0 0 0 1
y
M
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Transformations
Normal vectors transform differently than do ordinary points and directions A normal vector represents the direction
pointing out of a surface A normal vector is perpendicular to the
tangent plane If a matrix M transforms points from one
coordinate system to another, then normal vectors must be transformed by (M-1)T
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Geometry
A line in 3D space is represented by
S is a point on the line, and V is the direction along which the line runs
Any point P on the line corresponds to a value of the parameter t
Two lines are parallel if their direction vectors are parallel
t t P S V
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Geometry
A plane in 3D space can be defined by a normal direction N and a point P
Other points in the plane satisfy
PQ
N
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Geometry
A plane equation is commonly written
A, B, and C are the components of the normal direction N, and D is given by
for any point P in the plane
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Geometry
A plane is often represented by the 4D vector (A, B, C, D)
If a 4D homogeneous point P lies in the plane, then (A, B, C, D) P = 0
If a point does not lie in the plane, then the dot product tells us which side of the plane the point lies on
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Geometry
Distance d from a point P to a lineS + t V
P
VS
d
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Geometry
Use Pythagorean theorem:
Taking square root,
If V is unit length, then V 2 = 1
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Geometry
Intersection of a line and a plane Let P(t) = S + t V be the line Let L = (N, D) be the plane We want to find t such that L P(t) = 0
Careful, S has w-coordinate of 1, and V has w-coordinate of 0
x x y y z z w
x x y y z z
L S L S L S Lt
L V L V L V
L S
L V
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Geometry
If L V = 0, the line is parallel to the plane and no intersection occurs
Otherwise, the point of intersection is t
L S
P S VL V