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2 d transformations

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Prof. Dr. K. Adisesha
Prof. Dr. K. Adisesha

The document discusses 2D transformations in computer graphics, including translation, rotation, scaling, and shearing. Translation moves an object by adding offsets to x and y coordinates. Rotation rotates objects around the origin by applying trigonometric functions to x and y. Scaling enlarges or shrinks objects along the x- and y-axes. Shearing distorts objects along an axis based on their position on the other axis. Homogeneous coordinates allow transformations like translation, rotation, and scaling to be expressed using matrix multiplication.

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Computer Graphics 2D Transformations A transformation is any operation on a point in space (x, y) that maps the point's coordinates into a new set of coordinates (x1,y1). Translation In translation an object is displaced a given distance and direction from its original position. If the displacement is given by the vector v = txI + tyJ, the new object point P'(x', y') can be found by applying the transformation Tv to P(x, y). See the figure below P' = Tv(P) where x' = x + tx and y' = y + ty. As an example, consider a triangle defined by three vertices (20,0), (60, 0), and (40, 100) being translated 100 units to the right along the x-axis ( tx = 100) and 10 units up along the y-axis (ty = 10). The new vertices are (120, 10), (160, 10), and (140, 110), see figure below: Rotation In rotation, the object is rotated ø° about the origin. The convention is that the direction of the rotation is CCW if ø is a positive angle and CW if the ø is a negative angle. The transformation for rotation Rø is P' = Rø(P) where x' = x cos(ø) - y sin(ø) and y' = x sin(ø) + y cos(ø) For example a triangle (20,0), (60,0), (40,100) rotated 45° clockwise about the origin is (14.14, -14.14), (42.43, -42.43), (98.99, -42.43) K. Adisesha 1
Computer Graphics Scaling Scaling is the process of expanding or compressing the dimensions of an object. Positive scaling constants Sx and Sy are used to describe changes in length with respect to the x direction and y direction. A scaling constant > 1 creates an expansion (magnification) of length, and < 1 a compression (reduction) of length. Scaling occurs along the x-axis and y-axis to create a new point from the original. This is achieved using the following transformation: P' = TSx,Sy (P), where x' = Sx * x ,and y' = Sy * y If Sx and Sy are not equal, they have the effect of distorting pictures by elongating or shrinking them along the directions parallel to the coordinate axes. The mirror image of an object can be generated by using the negative values for Sxand Sy. Homogeneous Coordinates Translation, scaling and rotation were expressed non-homogenously: P´ = P + T P´ = S × P P´ = R × P Composition is difficult to express using the standard notation above. Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 × 3 matrices. Add a third coordinate to a point P(x,y). So instead of representing the point using an (x,y) coordinate pair, each point is represented by three values, (x, y, W). P2d (x, y) -> Ph (wx, wy, w), w xb9 0 Given Ph (x, y, w), w xb9 0 Then P2d (x, y) = P2d (x/w, y/w) W is 1 for affine transformations in graphics. Affine transformations have the property of preserving parallism of lines, but not the lengths and angles. See example in figure 5.6 on page 207 in your Computer Graphics text. Shear an affine transformation A shear is a transformation that distorts the shape of an object along either or both of the axies. Like scale and translate, a shear can be done along just one or along both of the coordinate axes. A shear along one axis (say, the x-axis) is performed in terms of the point's coordinate in the other axis (the y-axis). Thus a shear of 1 in the x-axis will cause the x-coodinate of the point ot distort by 1*(y-coordinate). K. Adisesha 2
Computer Graphics To shear in the x direction the equation is: x1 = x + ay y1 = y Where b = 0 Where x1 and y1 are the new values, x and y are the original values, and a is the scaling factor in the x direction. The matrix is as follows. Shearing in the y direction is similar except the roles are reversed. x1 = x y1 = y + bx Where a = 0. Where x1 and y1 are the new values, x and y are the original values, and b is the scaling factor in the y direction. The matrix is as follows. Example Original Y-Shear X-Shear K. Adisesha 3
Computer Graphics Composition R(ø) rotates about the origin; to rotate about P1 Translate P1 to origin Rotate Translate origin back to P1 K. Adisesha 4

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2 d transformations

  • 1. Computer Graphics 2D Transformations A transformation is any operation on a point in space (x, y) that maps the point's coordinates into a new set of coordinates (x1,y1). Translation In translation an object is displaced a given distance and direction from its original position. If the displacement is given by the vector v = txI + tyJ, the new object point P'(x', y') can be found by applying the transformation Tv to P(x, y). See the figure below P' = Tv(P) where x' = x + tx and y' = y + ty. As an example, consider a triangle defined by three vertices (20,0), (60, 0), and (40, 100) being translated 100 units to the right along the x-axis ( tx = 100) and 10 units up along the y-axis (ty = 10). The new vertices are (120, 10), (160, 10), and (140, 110), see figure below: Rotation In rotation, the object is rotated ø° about the origin. The convention is that the direction of the rotation is CCW if ø is a positive angle and CW if the ø is a negative angle. The transformation for rotation Rø is P' = Rø(P) where x' = x cos(ø) - y sin(ø) and y' = x sin(ø) + y cos(ø) For example a triangle (20,0), (60,0), (40,100) rotated 45° clockwise about the origin is (14.14, -14.14), (42.43, -42.43), (98.99, -42.43) K. Adisesha 1
  • 2. Computer Graphics Scaling Scaling is the process of expanding or compressing the dimensions of an object. Positive scaling constants Sx and Sy are used to describe changes in length with respect to the x direction and y direction. A scaling constant > 1 creates an expansion (magnification) of length, and < 1 a compression (reduction) of length. Scaling occurs along the x-axis and y-axis to create a new point from the original. This is achieved using the following transformation: P' = TSx,Sy (P), where x' = Sx * x ,and y' = Sy * y If Sx and Sy are not equal, they have the effect of distorting pictures by elongating or shrinking them along the directions parallel to the coordinate axes. The mirror image of an object can be generated by using the negative values for Sxand Sy. Homogeneous Coordinates Translation, scaling and rotation were expressed non-homogenously: P´ = P + T P´ = S × P P´ = R × P Composition is difficult to express using the standard notation above. Homogeneous coordinates allow all three to be expressed homogeneously, using multiplication by 3 × 3 matrices. Add a third coordinate to a point P(x,y). So instead of representing the point using an (x,y) coordinate pair, each point is represented by three values, (x, y, W). P2d (x, y) -> Ph (wx, wy, w), w xb9 0 Given Ph (x, y, w), w xb9 0 Then P2d (x, y) = P2d (x/w, y/w) W is 1 for affine transformations in graphics. Affine transformations have the property of preserving parallism of lines, but not the lengths and angles. See example in figure 5.6 on page 207 in your Computer Graphics text. Shear an affine transformation A shear is a transformation that distorts the shape of an object along either or both of the axies. Like scale and translate, a shear can be done along just one or along both of the coordinate axes. A shear along one axis (say, the x-axis) is performed in terms of the point's coordinate in the other axis (the y-axis). Thus a shear of 1 in the x-axis will cause the x-coodinate of the point ot distort by 1*(y-coordinate). K. Adisesha 2
  • 3. Computer Graphics To shear in the x direction the equation is: x1 = x + ay y1 = y Where b = 0 Where x1 and y1 are the new values, x and y are the original values, and a is the scaling factor in the x direction. The matrix is as follows. Shearing in the y direction is similar except the roles are reversed. x1 = x y1 = y + bx Where a = 0. Where x1 and y1 are the new values, x and y are the original values, and b is the scaling factor in the y direction. The matrix is as follows. Example Original Y-Shear X-Shear K. Adisesha 3
  • 4. Computer Graphics Composition R(ø) rotates about the origin; to rotate about P1 Translate P1 to origin Rotate Translate origin back to P1 K. Adisesha 4