2D-transformation-1.pdf
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This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.
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Computer graphics - bresenham line drawing algorithm
You know that DDA algorithm is an incremental scan conversion method which performs calculations at each step using the results from the preceding step. Here we are going to discover an accurate and efficient raster line generating algorithm, the Bresenham's line-drawing algorithm.
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The function given below handles all lines and implements the complete Bresenham's algorithm.
function line(x0, x1, y0, y1)
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
int deltax := x1 - x0
int deltay := abs(y1 - y0)
real error := 0
real deltaerr := deltay / deltax
int y := y0
if y0 < y1 then ystep := 1 else ystep := -1
for x from x0 to x1
if steep then plot(y,x) else plot(x,y)
error := error + deltaerr
if error ? 0.5
y := y + ystep
error := error - 1.0
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Here are the steps to solve this question:
(a) Given:
Window: wxmin = 2, wymin = 2
wxmax = 3, wymax = 4
Viewport: vxmin = 0, vymin = 0
vxmax = 1, vymax = 1
Sx = 1 - 0 = 1/1 = 1
3 - 2 = 1
Sy = 1 - 0 = 1/2 = 1/2
4 - 2 = 2
N = 1 0 0
0 1 0
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(b) Given:
Window: wxmin = 2, wymin = 2
wxmax = 3, wymax = 4
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This algorithm was developed by Jack E. Bresenham in 1962 at IBM.
The function given below handles all lines and implements the complete Bresenham's algorithm.
function line(x0, x1, y0, y1)
boolean steep := abs(y1 - y0) > abs(x1 - x0)
if steep then
swap(x0, y0)
swap(x1, y1)
if x0 > x1 then
swap(x0, x1)
swap(y0, y1)
int deltax := x1 - x0
int deltay := abs(y1 - y0)
real error := 0
real deltaerr := deltay / deltax
int y := y0
if y0 < y1 then ystep := 1 else ystep := -1
for x from x0 to x1
if steep then plot(y,x) else plot(x,y)
error := error + deltaerr
if error ? 0.5
y := y + ystep
error := error - 1.0
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Here are the steps to solve this question:
(a) Given:
Window: wxmin = 2, wymin = 2
wxmax = 3, wymax = 4
Viewport: vxmin = 0, vymin = 0
vxmax = 1, vymax = 1
Sx = 1 - 0 = 1/1 = 1
3 - 2 = 1
Sy = 1 - 0 = 1/2 = 1/2
4 - 2 = 2
N = 1 0 0
0 1 0
0 0 1
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Window: wxmin = 2, wymin = 2
wxmax = 3, wymax = 4
Viewport
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This document discusses various 2D geometric transformations including translation, rotation, scaling, and more complex transformations. Translation moves an object by adding offsets to x and y coordinates. Rotation repositions an object along a circular path defined by a rotation angle and pivot point. Scaling changes an object's size by multiplying x and y coordinates by scaling factors. More advanced topics covered include reflection, shear transformations, and performing multiple transformations sequentially as composites.
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Two-dimensional transformations include translations, rotations, and scalings. Transformations manipulate objects by altering their coordinate descriptions without redrawing them. Matrices can represent linear transformations and are used to describe 2D transformations. Common 2D transformations include translation by adding offsets to coordinates, rotation by applying a rotation matrix, and scaling by multiplying coordinates by scaling factors. More complex transformations can be achieved by combining basic transformations through matrix multiplication in a specific order.
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This document discusses different types of geometric transformations, focusing on translation. It defines translation as the movement of an object from one position to another without deformation. Translation is represented by adding offsets (tx, ty) to the original x and y coordinates. The offsets form the translation vector or shift vector (Tx, Ty). Translation of multiple points can be represented using matrix operations, where the translated point P' is equal to the original point P plus the translation vector T. Examples are provided to demonstrate translating single points and geometric shapes.
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it is related to Computer Graphics Subject.in this ppt we describe what is 2D Transformation, Translation, Rotation, Scaling : Uniform Scaling,Non-uniform Scaling ;Reflection,Shear,Composite Transformations
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This document provides an overview of 2D and 3D transformations using matrices. It discusses 2D rotation, translation, and scaling matrices. For 3D, it covers linear transformations, orthogonal matrices, handedness, cross products, and rotation matrices for the x, y, and z axes. It also discusses combining translation and rotation using 4x4 matrices, concatenating transformations, and how these concepts are applied in Direct3D.
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2D-transformation-1.pdf
- 2. 2D Transformations “Transformations are the operations applied to geometrical description of an object to change its position, orientation, or size are called geometric transformations”.
- 3. Translation Translation is a process of changing the position of an object in a straight-line path from one co- ordinate location to another. We can translate a two dimensional point by adding translation distances, tx and ty. Suppose the original position is (x ,y) then new position is (x’, y’). Here x’=x + tx and y’=y + ty.
- 4. Click icon to add picture
- 5. Translation
- 6. Matrix form of the equations: X’ = X + tx and Y’ = Y + ty is P = x P’ = x’ T= tx y y’ ty we can write it, P’= P + T
- 7. Translate a polygon with co-ordinates A(2,5) B(7,10) and C(10,2) by 3 units in X direction and 4 units in Y direction. A’ = A +T = 2 + 3 = 5 5 4 9 B’ = B + T = 7 + 3 = 10 10 4 14 C’ = C + T = 10 + 3 = 13 2 4 7
- 8. Rotation A two dimensional rotation is applied to an object by repositioning it along a circular path in the xy plane. Using standard trigonometric equations , we can express the transformed co-ordinates in terms of x’ = r cos( coscosr sinsin y’ = r sin( cosr sin The original co-ordinates of the point is x = r cos y = r sin
- 9. Click icon to add picture After substituting equation 2 in equation 1 we get x’=x cos Y’=x sin + y cos
- 10. Rotation
- 11. That equation can be represented in matrix form x’ y’ = x y cos -sin cos we can write this equation as, P’ = P . R Where R is a rotation matrix and it is given as R = cos -sin cos
- 12. A point (4,3) is rotated counterclockwise by angle of 45. find the rotation matrix and the resultant point. R = cos = cos45 sin45 - cos -sin45 cos45 = 1/√2 1/√2 - 1/√2 1/√2 P’ = 4 3 1/√2 1/√2 - 1/√2 1/√2 = 4/√2 – 3/√2 4/√2 + 3/√2 = 1/√2 7/√2
- 13. Scaling A scaling transformation changes the size of an object. This operation can be carried out for polygons by multiplying the co-ordinates values (x , y) of each vertex by scaling factors Sx and Sy to produce the transformed co-ordinates (x’ , y’). x’ = x . Sx y’ = y . Sy In the matrix form x’ y’ = x y Sx 0 0 Sy = P . S
- 14. Scaling • Uniform Scaling Un-uniform Scaling
- 15. Homogeneous co-ordinates for Translation The homogeneous co-ordinates for translation are given as T = 1 0 0 0 1 0 tx ty 1 Therefore , we have x’ y’ 1 = x y 1 1 0 0 0 1 0 tx ty 1 = x + tx y + ty 1
- 16. Homogeneous co-ordinates for rotation The homogeneous co-ordinates for rotation are given as R = cos sin -sin cos 0 0 1 Therefore , we have x’ y’ 1 = x y 1 cos sin -sin cos 0 0 1 = x cos - y sin x sin + y cos 1
- 17. Homogeneous co-ordinates for scaling The homogeneous co-ordinate for scaling are given as S = Sx 0 0 0 Sy 0 0 0 1 Therefore , we have x’ y’ 1 = x y 1 Sx 0 0 0 Sy 0 0 0 1 = x . Sx y . Sy 1
- 28. CONCLUSION To manipulate the initially created object and to display the modified object without having to redraw it, we use Transformations.
- 29. THANK YOU