Chapter 4.1 Mathematical Concepts. 2 Applied Trigonometry Trigonometric functions Defined using...

Chapter 4.1Mathematical Concepts

2

Applied Trigonometry

Trigonometric functions Defined using right triangle

x

yh

3


Angles measured in radians

Full circle contains 2 radians

4


Sine and cosine used to decompose a point into horizontal and vertical components

r cos

r sin r

x

y

5


Trigonometric identities

6


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

7


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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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 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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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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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 add and subtract componentwise

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The magnitude of an n-dimensional vector V is given by

In three dimensions, this is

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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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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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The entry of a matrix M in the i-th row and j-th column is denoted Mij

For example,

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The transpose of a matrix M is denoted MT and has its rows and columns exchanged:

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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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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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Example matrix product

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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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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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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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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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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

38

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

50

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

51

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

Chapter 4.1 Mathematical Concepts. 2 Applied Trigonometry Trigonometric functions Defined using...

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