Computer Graphics

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Dr.S.Sivakumar,Principal

C.P.A College, Bodinayakanur

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Transformations

In graphics, once we have an object described, transformations are used to move that object, scale it and rotate it

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Translation

Simply moves an object from one position to another

x_new = x_old + dx y_new = y_old + dy

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Rotation

Rotates all coordinates by a specified angle

• x_new = x_old × cosθ – y_old × sinθ
• y_new = x_old × sinθ + y_old × cosθ

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Points are always rotated about the origin

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2-D Rotation

x’ = x cos(θ) - y sin(θ)

y’ = x sin(θ) + y cos(θ)

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2-D Rotation

x = r cos (φ)

y = r sin (φ)

x’ = r cos (φ + θ)

y’ = r sin (φ + θ)

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Trig Identity…

x’ = r cos(φ) cos(θ) – r sin(φ) sin(θ)

y’ = r sin(φ) sin(θ) + r cos(φ) cos(θ)

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Substitute…

x’ = x cos(θ) - y sin(θ)

y’ = x sin(θ) + y cos(θ)

φ

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2-D Rotation

This is easy to capture in matrix form:

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Even though sin(θ) and cos(θ) are nonlinear functions of θ,

• x’ is a linear combination of x and y
• y’ is a linear combination of x and y

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Scaling

Scalar multiplies all coordinates

WATCH OUT: Objects grow and move!

x_new = Sx × x_old y_new = Sy × y_old

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Scaling

Scaling a coordinate means multiplying each of its components by a scalar

Uniform scaling means this scalar is the same for all components:

× 2

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Scaling

Non-uniform scaling: different scalars per component:

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How can we represent this in matrix form?

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Scaling

Scaling operation:

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Or, in matrix form:

scaling matrix

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Matrix Representation

Represent 2D transformation by a matrix��

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Multiply matrix by column vector� ⇔ apply transformation to point

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Matrix Representation

Transformations combined by multiplication

Matrices are a convenient and efficient way

to represent a sequence of transformations!

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2x2 Matrices

Transformations can be represented with a 2x2 matrix?

2D Identity?

2D Scale around (0,0)?

2D Rotate around (0,0)?

2D Shear?

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2x2 Matrices

Transformations can be represented with a 2x2 matrix?

2D Mirror about Y axis?

2D Mirror over (0,0)?

2D Translation?

NO!

Only linear 2D transformations

can be represented with a 2x2 matrix

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Basic 2D Transformations

• Translation:
• x’ = x + t_x
• y’ = y + t_y
• Scale:
• x’ = x * s_x
• y’ = y * s_y
• Shear:
• x’ = x + h_x*y
• y’ = y + h_y*x
• Rotation:
• x’ = x*cosθ - y*sinθ
• y’ = x*sinθ + y*cosθ

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Homogeneous Coordinates

Homogeneous coordinates

• represent coordinates in 2 dimensions with a 3-vector

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Homogeneous coordinates seem unintuitive, but they make graphics operations much easier

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Homogeneous Coordinates?

Mathematicians commonly use homogeneous coordinates as they allow scaling factors to be removed from equations

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The transformations we discussed previously can be represented as 3*3 matrices

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Using homogeneous coordinates allows us use matrix multiplication to calculate transformations – extremely efficient!

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Homogeneous Coordinates

• A point (x, y) can be re-written in homogeneous coordinates as (x_h, y_h, h)

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• The homogeneous parameter h is a non-�zero value such that:

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• We can then write any point (x, y) as (hx, hy, h)

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• We can conveniently choose h = 1 so that �(x, y) becomes (x, y, 1)

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Homogeneous Coordinates

• Add a 3rd coordinate to every 2D point
• (x, y, w) represents a point at location (x/w, y/w)
• (x, y, 0) represents a point at infinity
• (0, 0, 0) is not allowed

Convenient coordinate system to represent many useful transformations

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Homogeneous Coordinates

Represent translation as a 3x3 matrix?

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Using the rightmost column:

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Translation

Example of Translation

• α

t_x = 2�t_y = 1

Homogeneous Coordinates

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Basic 2D Transformations

Basic 2D transformations as 3x3 homogeneous matrices

Translate

Rotate

Shear

Scale

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Remember Matrix Multiplication

Recall how matrix multiplication takes place:

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Homogeneous Translation

The translation of a point by (dx, dy) can be written in matrix form as:

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Representing the point as a homogeneous column vector we perform the calculation as:

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Homogenous Transformations

To make operations easier, 2-D points are written as homogenous coordinate column vectors

Translation:

Scaling:

Rotation:

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Inverse Transformations

Transformations can easily be reversed using inverse transformations

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Combining Transformations

A number of transformations can be combined into one matrix to make things easy

• Allowed by the fact that we use homogenous coordinates

Rotating a polygon around a point other than the origin

• Transform to centre point to origin
• Rotate around origin
• Transform back to centre point

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Combining Transformations (cont.)

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Combining Transformations (cont.)

The three transformation matrices are combined as follows

Remember: Matrix multiplication is not commutative

so order matters

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Matrix Composition

Transformations can be combined by �matrix multiplication

p’ = T(t_x,t_y) R(Θ) S(s_x,s_y) p

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Matrix Composition

• Matrices are a convenient and efficient way to represent a sequence of transformations
• General purpose representation
• Hardware matrix multiply

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p’ = (T * (R * (S*p) ) )

p’ = (T*R*S) * p

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Matrix Composition

• After correctly ordering the matrices
• Multiply matrices together
• What results is one matrix – store it (on stack)!
• Multiply this matrix by the vector of each vertex
• All vertices easily transformed with one matrix multiply

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