Computer Graphics

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Dr.S.Sivakumar,Principal

C.P.A College, Bodinayakanur

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Two-Dimensional Geometric Transformations�

Two-Dimensional Geometric Transformations

• Basic Transformations
• Translation
• Rotation
• Scaling
• Composite Transformations
• Other transformations
• Reflection
• Shear

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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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Translation

• Translation transformation

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• Translation vector or shift vector T = (t_x, t_y)
• Rigid-body transformation
• Moves objects without deformation

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x

y

p

P’

T

x

y

T

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

y

x

0

1

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2

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9

10

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5

6

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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

x’ = x cos(θ) - y sin(θ)

y’ = x sin(θ) + y cos(θ)

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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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Rotation

• Rotation transformation

x

y

P(x,y)

P’ (x’,y’)

r

θ Φ

x’=rcos(Φ+θ)= rcos Φ cos θ -rsin Φ sin θ

y’=rsin(Φ+θ)= rcos Φ sin θ+rsin Φ cos θ

x=rcos Φ y=rsin Φ

x’=x cos θ -ysin θ

y’=xsin θ +ycos θ

P’= R· P

Rotation

• Pivot point

x

y

P(x,y)

P’ (x’,y’)

r

(x_r,y_r)

x’=x_r+(x- x_r)cos θ -(y- y_r)sin θ

y’=y_r+(x- x_r)sin θ +(y- y_r)cos θ

θ Φ

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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Scaling

• Scaling transformation

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• Scaling factors, s_x and s_y
• Uniform scaling

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x

y

x

y

Scaling

• Fixed point

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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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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Matrix Representations and Homogeneous Coordinates

• Homogeneous Coordinates

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• Matrix representations
• Translation

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Matrix Representations

• Matrix representations
• Scaling

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• Rotation

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Composite Transformations

• Translations

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Composite Transformations

• Scaling

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Composite Transformations

• Rotations

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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.)

1

2

3

4

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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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General Pivot-Point Rotation

• Rotations about any selected pivot point (x_r,y_r)
• Translate-rotate-translate

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General Pivot-Point Rotation

General Fixed-Point Scaling

Scaling with respect to a selected fixed position (x_f,y_f)

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General Fixed-Point Scaling

• Translate-scale-translate

General Scaling Directions

• Scaling factors s_x and s_y scale objects along the x and y directions.
• We scale an object in other directions with scaling factors s₁ and s₂

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General Scaling Directions

Concatenation Properties

• Matrix multiplication is associative.

A·B ·C = (A·B )·C = A·(B ·C)

• Transformation products may not be commutative
• Be careful about the order in which the composite matrix is evaluated.
• Except for some special cases:
• Two successive rotations
• Two successive translations
• Two successive scalings
• rotation and uniform scaling

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Concatenation Properties

• Reversing the order
• A sequence of transformations is performed may affect the transformed position of an object.

General Composite Transformations and Computer Efficiency

• A general two-dimensional transformation

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• Rotation-scaling terms rs_ij
• Translational terms trs_x and trs_y
• Minimum number of computations
• Four multiplications
• Four additions

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Rigid-Body Transformation

• Rigid-body transformation matrix

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• The upper-left 2-by-2 submatrix is an orthogonal matrix
• Two vectors (r_xx, r_xy) and (r_yx, r_yy) form an orthogonal set of unit vectors.

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• Multiplicative rotation terms r_ij
• Translational terms tr_x and tr_y

Rigid-Body Transformation

• The orthogonal property of rotation matrices
• We know the final orientation of an object

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Construct the desired transformation by assigning the elements of u’ to the first row of the rotation matrix and the elements of v’ to the second row.

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Computational Efficiency

• Use approximations and iterative calculations to reduce computations
• Approximate the trigonometric functions based on the first few terms of their power-series expansions.
• For small enough angles (< 10⁰), cosθ is approximately 1.sinθ is approximately θ
• Accumulated error control
• Estimate the error in x’ and y’ at each step
• Reset object positions when the error accumulation becomes too great

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Reflection

• A transformation produces a mirror image of an object.
• Axis of reflection
• A line in the xy plane
• A line perpendicular to the xy plane
• The mirror image is obtained by rotating the object 180⁰ about the reflection axis.
• Rotation path
• Axis in xy plane: in a plane perpendicular to the xy plane.
• Axis perpendicular to xy plane: in the xy plane.

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Reflection

• Reflection about the x axis

Reflection

• Reflection about the y axis

Reflection

• Reflection relative to the coordinate origin

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Reflection

• Reflection of an object relative to an axis perpendicular to the xy plane through P_rfl

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Reflection

• Reflection about the line y = x

Shear

• The x-direction shear relative to x axis

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If sh_x = 2:

Shear

• The x-direction shear relative to y = yref

If sh_x = ½ y_ref = -1:

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1/2

3/2

Shear

• The y-direction shear relative to x = x_ref

If sh_y = ½ x_ref = -1:

1

1/2

3/2

Transformations between Coordinate Systems

x

y

y’

x’

x₀

y₀

_θ

x

y

y’

x’

_θ

Transformations between Coordinate Systems

Method 1:

Method 2:

Affine Transformations

• A coordinate transformation of the form

x’=a_xxx+a_xyy+b_x, y’=a_yxx+a_yyy+b_y

• x’ and y’ is a linear function of the original coordinates x and y.
• a_ij and b_k are constants determined by the transformation type.
• Translation, rotation, scaling, reflection, and shear are two-dimensional affine transformations.
• An affine transformation involving only rotation, translation, and reflection preserves angles and lengths.

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