1. Savitribai Phule Pune University
Second Year of Mechanical Engineering (2019 Course)
Course Name:- Solid Modeling and Drafting(202042)
Unit No. IV : Geometric Transformation
Prepared by,
Mr. Hemant B Wagh
Assistant Professor
Department of Mechanical Engineering
Matoshri College of Engineering and Research Centre, Eklahare, Nashik
Matoshri College of Engineering and Research Centre, Nashik 1
2. Unit-IV: Geometric Transformation
• Syllabus
Transformations (2D & 3D) : Introduction, Formulation, Translation, Shear,
Rotation, Scaling and reflection,
Homogeneous representation,
Concatenated transformation,
Mapping of geometric models,
Inverse transformations,
Projections : Orthographic, Isometric, Perspective projections
Matoshri College of Engineering and Research Centre, Nashik 2
3. 2-Dimentional Geometric Transformations
Geometric transformations change the orientation, size & shape of the objects
in the database and of graphical image as well.
Where to use?
Constructing the model
Editing the model using commands like
Translate (Move), Reflect (Mirror), zoom,
View model from different positions.
Matoshri College of Engineering and Research Centre, Nashik 3
4. Geometric transformations
Geometric transformation: changes the orientation, size, and shape of the
objects in the database as well as on the graphics image.
Use of Geometric Transformation
In a construction of a model
In editing the model using the commands like : translate, rotate, zoom, mirror,
array, etc.;
For obtaining orthographic, isometric and prospective views of the model;
To view the model from different positions;
In animations. Matoshri College of Engineering and Research Centre, Nashik 4
5. Formulation of Geometric Transformations
Any 2D graphic element is constructed using points. Hence, transformation
of any element essentially means the transformation of all the points of the
element.
- Matrix representation of a Point P (x, y),
- 𝑃 =
𝑥
𝑦 ,
- Matrix representation of transformed point
P’ (x’, y’),
- 𝑃′ =
𝑥′
𝑦′
, ∴ 𝑃′ = 𝑇𝑅
𝑥
𝑦
- where 𝑇𝑅 = transformation matrixMatoshri College of Engineering and Research Centre, Nashik 5
6. Basic geometric transformations
1. Translation (Move)
2. Rotation
3. Scaling (Zoom)
4. Reflection (Mirror)
5. Shear
6. Concatenated (Composite or Combined) Transformation
Matoshri College of Engineering and Research Centre, Nashik 6
7. 1.Translation (Move)
- It involves moving the graphic element or object from one location to
other.
- Consider a point P (x, y) is translated by a distance 𝑡 𝑥 in X-direction
and 𝑡 𝑦 in Y-direction to a new position as shown. Therefore,
- 𝑥′ = 𝑥 + 𝑡 𝑥&𝑦′ = 𝑦 + 𝑡 𝑦.
- Then the equation in matrix form is,
-
𝑥′
𝑦′
=
𝑥
𝑦 +
𝑡 𝑥
𝑡 𝑦
- or 𝑃′ = 𝑃 + 𝑇 ,
Matoshri College of Engineering and Research Centre, Nashik 7
8. Translation (Move)
- where 𝑃′ =
𝑥′
𝑦′
= new position matrix,
- 𝑃 =
𝑥
𝑦 = original position matrix
- and 𝑇 =
𝑡 𝑥
𝑡 𝑦
= translation matrix
Matoshri College of Engineering and Research Centre, Nashik 8
9. 2.Rotation
- Involves rotating the graphic element from one location to other about
origin (Z-axis) by an angle 𝜃.
- Consider a point P (x, y) is rotated about Z-axis by an angle 𝜃 in
counterclockwise direction to a new position P’ (x’, y’) as shown.
- Let, r = constant distance of the point from the origin,
- ∅ = original angular position of the point with horizontal,
- 𝑥 = 𝑟 cos ∅
- &y = 𝑟 𝑠𝑖𝑛 ∅,
- 𝑥′ = 𝑟 cos ∅ + 𝜃 = 𝑟𝑐𝑜𝑠∅. 𝑐𝑜𝑠 𝜃 − 𝑟𝑠𝑖𝑛∅. 𝑠𝑖𝑛𝜃&
- 𝑦′
= 𝑟 sin ∅ + 𝜃 = 𝑟𝑠𝑖𝑛∅. 𝑐𝑜𝑠 𝜃 + 𝑟𝑠𝑖𝑛∅. 𝑐𝑜𝑠𝜃,
- 𝑥′
= 𝑟𝑐𝑜𝑠∅. 𝑐𝑜𝑠𝜃 − 𝑟𝑠𝑖𝑛∅. 𝑠𝑖𝑛𝜃
Matoshri College of Engineering and Research Centre, Nashik 9
10. Rotation
- & 𝑦′ = 𝑟𝑐𝑜𝑠∅. 𝑠𝑖𝑛𝜃 + 𝑟𝑠𝑖𝑛∅. 𝑐𝑜𝑠𝜃
-
𝑥′
𝑦′
=
cos 𝜃 −𝑠𝑖𝑛𝜃
sin 𝜃 cos 𝜃
𝑥
𝑦
- or 𝑃′ = 𝑅 𝑃 ,
- where 𝑃′ =
𝑥′
𝑦′
= New position matrix,
- 𝑃 =
𝑥
𝑦 = Original position matrix
- and 𝑅 =
cos 𝜃 −𝑠𝑖𝑛𝜃
sin 𝜃 cos 𝜃
= Rotation Matrix
Matoshri College of Engineering and Research Centre, Nashik 10
11. 3.Scaling (Zoom)
- A scaling alters the size of the graphics element or object. It is used to
enlarge or reduce the size of the element.
- Let, 𝑆 𝑥 = Scaling factors in X-direction,
- 𝑆 𝑦 = Scaling factors in Y-direction
- ∴ 𝑥′
= 𝑥. 𝑆 𝑥&𝑦′
= 𝑦. 𝑆 𝑦,
- The equation can be written,
-
𝑥′
𝑦′
=
𝑆 𝑥 0
0 𝑆 𝑦
𝑥
𝑦 or 𝑃′ = 𝑆 𝑃 ,
- where, 𝑆 = 𝑆𝑐𝑎𝑙𝑖𝑛𝑔 𝑀𝑎𝑡𝑟𝑖𝑥 =
𝑆 𝑥 0
0 𝑆 𝑦Matoshri College of Engineering and Research Centre, Nashik 11
12. 3.Scaling (Zoom)
Any positive value can be used as scaling factor
scaling factor < 1 reduce the size of the object
scaling factor > 1 enlarge the object
scaling factor = 1 then the object stays unchanged
If Sx = Sy , we call it uniform scaling
If Sx ≠ Sy , we call it non-uniform scaling
Matoshri College of Engineering and Research Centre, Nashik 12
13. 4.Reflection (Mirror)
It’s the transformation that produces a mirror image of the graphics element
about any axis or line.
Various commonly used reflections:
1. Reflection about X-axis
2. Reflection about Y-axis
3. Reflection about origin
4. Reflection about line y = x
5. Reflection about y = -xMatoshri College of Engineering and Research Centre, Nashik 13
14. Reflection (Mirror)
1. Reflection about X-axis:
- Consider a point P(x, y) is reflected about X-axis to a new position
P’(x’, y’). Then,
- 𝑥′ = 𝑥 and 𝑦′ = −𝑦,
- ∴
𝑥′
𝑦′
=
1 0
0 −1
𝑥
𝑦
- or 𝑃′ = 𝑀 𝑥 𝑃
- where, 𝑀 𝑥 = 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑏𝑜𝑢𝑡 𝑋 − 𝑎𝑥𝑖𝑠 =
1 0
0 −1Matoshri College of Engineering and Research Centre, Nashik 14
15. Reflection (Mirror)
2. Reflection about Y-axis:
- Consider a point P(x, y) is reflected about Y-axis to a new position
P’(x’, y’). Then,
- 𝑥′ = −𝑥 and 𝑦′ = 𝑦,
- ∴
𝑥′
𝑦′
=
−1 0
0 1
𝑥
𝑦
- or 𝑃′ = 𝑀 𝑦 𝑃
- where, 𝑀 𝑦 = 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑏𝑜𝑢𝑡 𝑌 − 𝑎𝑥𝑖𝑠 =
−1 0
0 1Matoshri College of Engineering and Research Centre, Nashik 15
16. Reflection (Mirror)
3. Reflection about Origin:
- Consider a point P(x, y) is reflected about the origin to a new position
P’(x’, y’). Then,
- 𝑥′ = −𝑥 and 𝑦′ = −𝑦,
- ∴
𝑥′
𝑦′
=
−1 0
0 −1
𝑥
𝑦
- or 𝑃′ = 𝑀 𝑜 𝑃
- where, 𝑀 𝑜 = 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑏𝑜𝑢𝑡 𝑜𝑟𝑖𝑔𝑖𝑛 =
−1 0
0 −1Matoshri College of Engineering and Research Centre, Nashik 16
17. Reflection (Mirror)
4. Reflection about line y = x:
- Consider a point P(x, y) is reflected about the line y = x to a new
position P’(x’, y’). Then,
- 𝑥′ = 𝑦 and 𝑦′ = 𝑥,
- ∴
𝑥′
𝑦′
=
0 1
1 0
𝑥
𝑦
- or 𝑃′ = 𝑀 𝑦𝑥 𝑃
- where, 𝑀 𝑦𝑥 = 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑏𝑜𝑢𝑡 𝑙𝑖𝑛𝑒 (𝑦 = 𝑥) =
0 1
1 0Matoshri College of Engineering and Research Centre, Nashik 17
18. Reflection (Mirror)
4. Reflection about line y = -x:
- Consider a point P(x, y) is reflected about the line y = - x to a new
position P’(x’, y’). Then,
- 𝑥′ = −𝑦 and 𝑦′ = −𝑥,
- ∴
𝑥′
𝑦′
=
0 −1
−1 0
𝑥
𝑦
- or 𝑃′ = 𝑀 𝑦−𝑥 𝑃
- where, 𝑀 𝑦−𝑥 = 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑚𝑎𝑡𝑟𝑖𝑥 𝑎𝑏𝑜𝑢𝑡 𝑙𝑖𝑛𝑒 (𝑦 = −𝑥) =
0 −1
−1 0Matoshri College of Engineering and Research Centre, Nashik 18
19. 5.Shear
This transformation distorts the shape of the graphics element such that the
distorted shape appears as if the element is composed of internal layers that
had been caused to slide over each other.
Types of Shear:
1. X-Direction Shear
2. Y-Direction Shear
Matoshri College of Engineering and Research Centre, Nashik 19
20. Shear
1. X-Direction Shear
- Consider a point P (x, y) is subjected to shear in X-direction so as to
occupy a new position P’ (x’, y’).
- Then, 𝑥′ = 𝑥 + 𝑆ℎ 𝑥. 𝑦 and 𝑦′ = 𝑦
- 𝑤ℎ𝑒𝑟𝑒, 𝑆ℎ 𝑥 = 𝑆ℎ𝑒𝑎𝑟 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑋 − 𝑑𝑖𝑟
- ∴
𝑥′
𝑦′
=
1 𝑆ℎ 𝑥
0 1
𝑥
𝑦 or 𝑃′ = 𝑆ℎ 𝑥 𝑃 ,
- where, 𝑆ℎ 𝑥 =
1 𝑆ℎ 𝑥
0 1
= 𝑋 − 𝑑𝑖𝑟 𝑆ℎ𝑒𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥
Matoshri College of Engineering and Research Centre, Nashik 20
21. Shear
2. Y-Direction Shear
- Consider a point P (x, y) is subjected to shear in Y-direction so as to occupy
a new position P’ (x’, y’).
- Then, 𝑥′ = 𝑥 and 𝑦′ = 𝑦 + 𝑆ℎ 𝑦. 𝑥
- 𝑤ℎ𝑒𝑟𝑒, 𝑆ℎ 𝑦 = 𝑆ℎ𝑒𝑎𝑟 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑌 − 𝑑𝑖𝑟
- ∴
𝑥′
𝑦′
=
1 0
𝑆ℎ 𝑦 1
𝑥
𝑦 or 𝑃′ = 𝑆ℎ 𝑦 𝑃 ,
- where, 𝑆ℎ 𝑦 =
1 0
𝑆ℎ 𝑦 1 = 𝑌 − 𝑑𝑖𝑟 𝑆ℎ𝑒𝑎𝑟 𝑚𝑎𝑡𝑟𝑖𝑥
Matoshri College of Engineering and Research Centre, Nashik 21
22. 6.Concatenated (Composite) Matrix
- It’s the combination of two or more transformations like translation,
rotation, scaling and reflection. A single transformation matrix known
as Concatenated Matrix or Composite Matrix.
- This combination of transformations into single matrix improves the
performance of graphics packages.
- Example – Consider a point P (x, y) is to be reoriented to a new
position P’ (x’, y’) by rotating first and then scaling, then concatenated
transformation can be given as follows:
- 𝑷′ = 𝑺 𝑹 𝑷
- or 𝑷′ = 𝑪𝑻 𝑷
Matoshri College of Engineering and Research Centre, Nashik 22
23. Concatenated (Composite) Matrix
- where 𝑅 = Rotation Matrix, 𝑆 = Scaling Matrix,
- 𝐶𝑇 = Concatenated Transformation Matrix and is given by,
- 𝐶𝑇 = 𝑆 𝑅
- =
𝑆 𝑥 0
0 𝑆 𝑦
cos 𝜃 − sin 𝜃
sin 𝜃 cos 𝜃
- =
𝑆 𝑥. cos 𝜃 −𝑆 𝑥. sin 𝜃
𝑆 𝑦. sin 𝜃 𝑆 𝑦. cos 𝜃
- ∴ 𝑪𝑻 =
𝑺 𝒙. 𝒄𝒐𝒔 𝜽 −𝑺 𝒙. 𝒔𝒊𝒏 𝜽
𝑺 𝒚. 𝒔𝒊𝒏 𝜽 𝑺 𝒚. 𝒄𝒐𝒔 𝜽 = 𝑪𝒐𝒏𝒄𝒂𝒕𝒆𝒏𝒂𝒕𝒆𝒅 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 23
24. 2D Geometric Transformations using Homogeneous
Coordinates
Need of homogeneous coordinates:
Composite matrix are obtained by combining two or more transformations together.
Some transformations like translate use addition and some use multiplication.
This makes it inconvenient to concatenate(composite) transformations.
Hence, in order to make it convenient for concatenation, it is desirable to express all
the transformations in the form of matrix multiplication only.
This can be effectively achieved by representing the points by their
homogenous coordinates.
Matoshri College of Engineering and Research Centre, Nashik 24
25. 2D Geometric Transformations using Homogeneous
Coordinates
- Homogeneous Coordinates: In this representation, a point in n-
dimensional space is represented by ‘n+1’ coordinates.
- So in two-dimensional space, point ‘P’ with cartesian coordinates (𝒙, 𝒚)
can be represented by homogeneous coordinates (𝒙 𝒉, 𝒚 𝒉, 𝒉), where, h
is any non-zero scalar factor. The homogeneous coordinates are
expressed in terms of two-dimensional cartesian coordinates as:
- 𝒙 𝒉 = 𝒉. 𝒙 and 𝒚 𝒉 = 𝒉. 𝒚
- The convenient value of ‘h’ is taken as 1. Therefore, any 2-D point P
(x, y) is expressed in homogeneous coordinate system as P (x, y, 1).
Matoshri College of Engineering and Research Centre, Nashik 25
26. Advantages of Homogeneous Coordinates
All transformations can be represented as 3*3 matrices making
homogeneity in representation.
Homogeneous representation allows us to use matrix multiplication to
calculate transformations extremely efficient.
Entire object transformation reduces to single matrix multiplication
operation.
Combined transformation are easier to built and understand.
Matoshri College of Engineering and Research Centre, Nashik 26
27. 2D Geometric Transformations using Homogeneous
Coordinates
- Hence, the 2 × 2 transformation matrices in 2-D cartesian coordinate
system are modified to 3 × 3 transformation matrices in homogeneous
coordinate system.
- Therefore, 2-D transformation matrix,
- 𝑻 =
𝒂 𝒃
𝒄 𝒅
can be written in homogeneous coordinate system as,
- 𝑻 𝒉 =
𝒂 𝒃 𝟎
𝒄 𝒅 𝟎
𝟎 𝟎 𝟏
Matoshri College of Engineering and Research Centre, Nashik 27
28. 1. Translation
-
𝑥′
𝑦′
=
𝑥
𝑦 +
𝑡 𝑥
𝑡 𝑦
can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
1 0 𝑡 𝑥
0 1 𝑡 𝑦
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑇ℎ 𝑃 ,
- where, 𝑻 𝒉 𝒐𝒓 𝑻 =
𝟏 𝟎 𝒕 𝒙
𝟎 𝟏 𝒕 𝒚
𝟎 𝟎 𝟏
= 𝑻𝒓𝒂𝒏𝒔𝒍𝒂𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 28
29. 2. Rotation
-
𝑥′
𝑦′
=
cos 𝜃 − sin 𝜃
sin 𝜃 cos 𝜃
𝑥
𝑦 can be expressed in homogeneous
coordinate system as,
-
𝑥′
𝑦′
1
=
cos 𝜃 − sin 𝜃 0
sin 𝜃 cos 𝜃 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑅 𝑃 ,
- where, 𝑹 =
𝐜𝒐𝒔 𝜽 − 𝒔𝒊𝒏 𝜽 𝟎
𝒔𝒊𝒏 𝜽 𝐜𝒐𝒔 𝜽 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒐𝒕𝒂𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 29
30. 3. Scaling
-
𝑥′
𝑦′
=
𝑆 𝑥 0
0 𝑆 𝑦
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
𝑆 𝑥 0 0
0 𝑆 𝑦 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑆 𝑃 ,
- where, 𝑺 =
𝑺 𝒙 𝟎 𝟎
𝟎 𝑺 𝒚 𝟎
𝟎 𝟎 𝟏
= 𝑺𝒄𝒂𝒍𝒊𝒏𝒈 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 30
31. 4. (a) Reflection about X-Axis
-
𝑥′
𝑦′
=
1 0
0 −1
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
1 0 0
0 −1 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑀 𝑥 𝑃 ,
- where, 𝑴 𝒙 =
𝟏 𝟎 𝟎
𝟎 −𝟏 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 𝒂𝒃𝒐𝒖𝒕 𝑿 − 𝑨𝒙𝒊𝒔
Matoshri College of Engineering and Research Centre, Nashik 31
32. 4. (b) Reflection about Y-Axis
-
𝑥′
𝑦′
=
−1 0
0 1
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
−1 0 0
0 1 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑀 𝑦 𝑃 ,
- where, 𝑴 𝒚 =
−𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 𝒂𝒃𝒐𝒖𝒕 𝒀 − 𝑨𝒙𝒊𝒔
Matoshri College of Engineering and Research Centre, Nashik 32
33. 4. (c) Reflection about Origin
-
𝑥′
𝑦′
=
−1 0
0 −1
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
−1 0 0
0 −1 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑀 𝑜 𝑃 ,
- where, 𝑴 𝒐 =
−𝟏 𝟎 𝟎
𝟎 −𝟏 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 𝒂𝒃𝒐𝒖𝒕 𝑶𝒓𝒊𝒈𝒊𝒏
Matoshri College of Engineering and Research Centre, Nashik 33
34. 4. (d) Reflection about line y = x
-
𝑥′
𝑦′
=
0 1
1 0
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
0 1 0
1 0 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑀 𝑦𝑥 𝑃 ,
- where, 𝑴 𝒚𝒙 =
𝟎 𝟏 𝟎
𝟏 𝟎 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 𝒂𝒃𝒐𝒖𝒕 𝒍𝒊𝒏𝒆 𝒚 = 𝒙
Matoshri College of Engineering and Research Centre, Nashik 34
35. 4. (e) Reflection about line y = -x
-
𝑥′
𝑦′
=
0 −1
−1 0
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
0 −1 0
−1 0 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑀 𝑦−𝑥 𝑃 ,
- Where,
- 𝑴 𝒚−𝒙 =
𝟎 −𝟏 𝟎
−𝟏 𝟎 𝟎
𝟎 𝟎 𝟏
= 𝑹𝒆𝒇𝒍𝒆𝒄𝒕𝒊𝒐𝒏 𝑴𝒂𝒕𝒓𝒊𝒙 𝒂𝒃𝒐𝒖𝒕 𝒍𝒊𝒏𝒆 𝒚 = −𝒙
Matoshri College of Engineering and Research Centre, Nashik 35
36. 5. (a) Shear in X-Direction
-
𝑥′
𝑦′
=
1 𝑆ℎ 𝑥
0 1
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
1 𝑆ℎ 𝑥 0
0 1 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑆ℎ 𝑥 𝑃 ,
- Where,
- 𝑺𝒉 𝒙 =
𝟏 𝑺𝒉 𝒙 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
= 𝑿 − 𝑫𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝑺𝒉𝒆𝒂𝒓 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 36
37. 5. (b) Shear in Y-Direction
-
𝑥′
𝑦′
=
1 0
𝑆ℎ 𝑦 1
𝑥
𝑦 can be expressed in homogeneous coordinate
system as,
-
𝑥′
𝑦′
1
=
1 0 0
𝑆ℎ 𝑦 1 0
0 0 1
𝑥
𝑦
1
or 𝑃′ = 𝑆ℎ 𝑦 𝑃 ,
- Where,
- 𝑺𝒉 𝒚 =
𝟏 𝟎 𝟎
𝑺𝒉 𝒚 𝟏 𝟎
𝟎 𝟎 𝟏
= 𝒀 − 𝑫𝒊𝒓𝒆𝒄𝒕𝒊𝒐𝒏 𝑺𝒉𝒆𝒂𝒓 𝑴𝒂𝒕𝒓𝒊𝒙
Matoshri College of Engineering and Research Centre, Nashik 37
38. 2-D Inverse Transformations
- Many a times, during the development of concatenated matrix, it is
necessary to use inverse of some basic geometric transformation
matrices.
- Let, 𝑃 =
𝑥
𝑦
1
= 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡,
- 𝑃′ =
𝑥′
𝑦′
1
= 𝑁𝑒𝑤 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑝𝑜𝑖𝑛𝑡, 𝑇𝑅 = 𝑇𝑟𝑎𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑀𝑎𝑡𝑟𝑖𝑥 &
- 𝑇𝑅 −1 = 𝐼𝑛𝑣𝑒𝑟𝑠𝑒 𝑇𝑟𝑎𝑛𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 𝑀𝑎𝑡𝑟𝑖𝑥
Matoshri College of Engineering and Research Centre, Nashik 38
40. 1. Inverse Translation
- Inverse Translation is obtained by replacing𝑡 𝑥 and𝑡 𝑦 by−𝑡 𝑥 and −𝑡 𝑦
respectively. Therefore, the inverse translation matrix is given by,
- 𝑇ℎ
−1
=
1 0 −𝑡 𝑥
0 1 −𝑡 𝑦
0 0 1
- 𝑯𝒆𝒏𝒄𝒆, 𝑷 = 𝑇ℎ
−𝟏
𝑷′
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41. 2. Inverse Rotation
- Inverse Rotation is obtained by replacing𝜃by −𝜃 in rotation matrix.
Therefore, the inverse rotation matrix is given by,
- 𝑅 −1 =
cos(−𝜃) −sin(−𝜃) 0
sin(−𝜃) cos(−𝜃) 0
0 0 1
- Or 𝑅 −1 =
cos 𝜃 sin 𝜃 0
−sin 𝜃 cos 𝜃 0
0 0 1
- 𝑯𝒆𝒏𝒄𝒆, 𝑷 = 𝑹 −𝟏 𝑷′
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42. 3. Inverse Scaling
- Inverse Scaling is obtained by replacing 𝑆 𝑥 and 𝑆 𝑦 by 1
𝑆 𝑥
and 1
𝑆 𝑦
respectively in Scaling matrix. Therefore, the inverse scaling matrix is
given by,
- 𝑆 −1 =
1
𝑆 𝑥
0 0
0 1
𝑆 𝑦
0
0 0 1
- 𝑯𝒆𝒏𝒄𝒆, 𝑷 = 𝑺 −𝟏 𝑷′
Matoshri College of Engineering and Research Centre, Nashik 42
43. 4. Inverse Reflection
- Inverse reflection matrices are same as the reflection matrices.
Therefore,
- 𝑴 𝒙
−𝟏 = 𝑴 𝒙
- 𝑴 𝒚
−𝟏
= 𝑴 𝒚
- 𝑴 𝒐
−𝟏 = 𝑴 𝒐
- 𝑴 𝒚𝒙
−𝟏
= 𝑴 𝒚𝒙
- 𝑴 𝒚−𝒙
−𝟏
= 𝑴 𝒚−𝒙 Matoshri College of Engineering and Research Centre, Nashik 43
44. Composite Transformation:
The number of operations will be reduced
1. Scaling about any point
2. Rotation about any point
3. Reflection about line y=mx+c
Matoshri College of Engineering and Research Centre, Nashik 44
45. 1.Scaling about any point
Suppose we want to perform scaling about an arbitrary point, then we
can perform it by the sequence of three transformations
• Translation
• Scaling
• Reverse Translation
𝑥′
𝑦′
1
=
𝑥
𝑦
1
[P’]=[𝑇]−1
[R][S][P]
Matoshri College of Engineering and Research Centre, Nashik 45
46. 2.Rotation about any point
Suppose we want to perform rotation about an arbitrary point, then we
can perform it by the sequence of three transformations
• Translation
• Rotation
• Reverse Translation
𝑥′
𝑦′
1
=
𝑥
𝑦
1
[P’]=[𝑇]−1
[R][T][P]
Matoshri College of Engineering and Research Centre, Nashik 46
47. 3.Reflection of line y=mx+c
can be accomplished with a combination of :
1. translate the Line so that it passes through the origin i.e., T(0, -c)
2. rotate the line so that it coincide with x axis
3. Reflect the give object about x axis.
4. restore the line to its original position with the inverse rotation
Matoshri College of Engineering and Research Centre, Nashik 47
48. 1. translate the Line so that it passes through the origin i.e., T(0, -c)
2. rotate the line so that it coincide with x axis
Matoshri College of Engineering and Research Centre, Nashik 48
49. 3. Reflect the give object about x axis.
4. restore the line to its original position with the inverse rotation
5. restore the line to its original position with the inverse translation.
Matoshri College of Engineering and Research Centre, Nashik 49
50. Reflection of line y=mx+c
Matoshri College of Engineering and Research Centre, Nashik 50
60. A triangle with vertices A(8,0),B(12,0) and C(12,3) has undergoes
the reflection about the line y=x. Find new coordinated of triangle
using transformation matrix.
• Solution:-
1. Mirror about line y=x
Matoshri College of Engineering and Research Centre, Nashik 60
63. New Coordinate of triangle
A’(0,8),B’(0,12),C’(3,12)
Matoshri College of Engineering and Research Centre, Nashik 63
64. An object is to be rotated about point A(-10,-10)by
90 degree CCW direction. Find Concatenated
transformation matrix.(Insem 6 marks)
Given
A(-10,-10) Angle=90 degree
Matoshri College of Engineering and Research Centre, Nashik 64
65. 1.Translation:
Translate the line such that point A(-10,-10)
coincides with origin. Hence translation distance
are
tx = 10 and ty=10
Matoshri College of Engineering and Research Centre, Nashik 65
66. 2.Rotation
Rotate the line about Z-axis(i.e. Origin) through 90 in
CCW direction.
Matoshri College of Engineering and Research Centre, Nashik 66
69. A line drawn between P1(2,4) and p2(6,8) is rotated
by 30 in CCW direction about point P1. Derive the
concatenated transformation matrix and find new
coordinated of line after transformation.(Insem 6
marks)
Solution
Concatenated transformation matrix
The rotation of line through 30 in CCW direction about point P1(2,4)
can be achieved in three steps
Matoshri College of Engineering and Research Centre, Nashik 69
76. Line A(5,5) B(10,15) is to be rotates about point
B by 600 in CCW. Find new position of point A
and B of line.(Dec-15 Endsem-6marks)
Solution
Given:- A(5,5) B(10,15) ө= 600
Matoshri College of Engineering and Research Centre, Nashik 76
77. ConcatenatedTransformation matrix:-
Rotation of line through 600 in CCW about point
B(10,15)
Translation
B(10,15) coincides hence translation distance are
tx = -10 and ty =-15
[Th] = =
Matoshri College of Engineering and Research Centre, Nashik 77
89. Mapping of geometric models
Types of coordinate system
1. Model Coordinate System(word CS/Global CS)
2. Local Coordinate System (User CS/ working CS)
Conversion form one coordinates system to another coordinate
system
Global CS to Local CS
Local CS to Global CS
Matoshri College of Engineering and Research Centre, Nashik 89
90. Types of coordinate systems
Matoshri College of Engineering and Research Centre, Nashik 90
91. Mapping of geometric models
Translation mapping
Rotational mapping
Matoshri College of Engineering and Research Centre, Nashik 91
92. Mapping of geometric models
Translation mapping- axes of coordinates parallel
𝑥′
𝑦′
𝑧′
1
=
1 0 0 − 𝑑𝑥
0 1 0 − 𝑑𝑦
0 0 1 − 𝑑𝑧
0 0 0 1
∗
𝑥
𝑦
𝑧
1
𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑙𝑜𝑐𝑎𝑙 𝐶𝑆
𝑇𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑎𝑝𝑝𝑖𝑛𝑔
𝑚𝑎𝑡𝑟𝑖𝑥
𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑔𝑙𝑜𝑏𝑎𝑙 𝐶𝑆
Matoshri College of Engineering and Research Centre, Nashik 92
93. Mapping of geometric models
Rotational mapping –two coordinate system common origin but
axes are some angle.
𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑙𝑜𝑐𝑎𝑙 𝐶𝑆
𝑅𝑜𝑡𝑎𝑡𝑖𝑜𝑛
𝑚𝑎𝑡𝑟𝑖𝑥
𝑝𝑜𝑖𝑛𝑡 𝑖𝑛 𝑔𝑙𝑜𝑏𝑎𝑙 𝐶𝑆
𝑥′
𝑦′
𝑧′
1
=
𝑐𝑜𝑠θ 𝑠𝑖𝑛θ 0 0
−𝑠𝑖𝑛θ 𝑐𝑜𝑠θ 0 0
0 0 1 0
0 0 0 1
∗
𝑥
𝑦
𝑧
1
Matoshri College of Engineering and Research Centre, Nashik 93
95. Parallel Projection
• Centre of the projection is taken at infinity
• Used in engineering drawings to generate orthographic view of the
object.
Matoshri College of Engineering and Research Centre, Nashik 95
96. Orthographic Projection
• Direction of projection is normal to the projection plane.
• There are three types of orthographic view −
1.Front view 2.Top view 3. Side view 1.Front view matrix
2.Top view matrix
3. Side view matrix
Matoshri College of Engineering and Research Centre, Nashik 96
97. Perspective Projection
• Centre of the projection is at finite distance
• Not suitable for engineering drawings
Matoshri College of Engineering and Research Centre, Nashik 97
99. Isometric projections
Isometric projections are commonly used in technical drawings
In an isometric projection the three axes appear 120° from each other
It can be achieved by rotating an object
i) 45° about z axis
ii) ~35.3° through the horizontal axis
Matoshri College of Engineering and Research Centre, Nashik 99
100. Isometric projections
i) 45° about z axis ii) ~35.3° through the horizontal
axis (Theta is 54.7°)
if we now multiply Rz(45°) by Rx(54.7°) we get
Matoshri College of Engineering and Research Centre, Nashik 100
101. ASSIGNMENT QUESTIONS:
1) Explain the geometric mapping and its types.
2) Write down various 3D homogeneous transformation matrix.
3) Differentiate between parallel and perspective projection.
4) Write a short note on orthographic projection and write matrix used to find out
various view of orthographies position.
5) Write a short note on perspective projection and derive the expression to find out
new coordinates.
6) Write a short note on isometric projections
Matoshri College of Engineering and Research Centre, Nashik 101
