Solid Modeling and Drafting - Unit 4 - Geometric Transformations PDF

SOLID MODELING AND DRAFTING Unit 4 – Geometric Transformation Content of Unit 4 Geometric Transformations Translation, Scaling, Rotation, Reflection/Mirror, Shear, Homogeneous Transformation, Inverse Transformation, Concatenated Transformation Coordinate systems Model (...

SOLID MODELING AND DRAFTING Unit 4 – Geometric Transformation Content of Unit 4 Geometric Transformations Translation, Scaling, Rotation, Reflection/Mirror, Shear, Homogeneous Transformation, Inverse Transformation, Concatenated Transformation Coordinate systems Model (MCS) coordinate system, Working (WCS) coordinate system, Screen (SCS) coordinate system, Mapping of coordinate systems Projections of geometric models Orthographic and Perspective projections, Design and Engineering applications By Prof. Swapnil Jadhav Dr. D. Y. Patil Institute of Technology, Pimpri, Pune Two-Dimensional Dimensional Geometric Transformations Geometric transformation means changing some graphics into something else by applying various types of transformations such as translation, scaling, rotation, shearing, reflection etc. When a transformation takes place on a 2D plane, it is called 2D transformation. ormation. Uses of Geometric transformation:  While creating geometric model  ii) In editing of geometric model with the help of different commands like translation, scaling, rotation, shearing, reflection etc.  iii) For obtaining different view of model (orthographic, isometric) Figure 1: Types of geometrical transformation Types of Geometric transformations  Translation  Scaling  Rotation  Shearing  Reflection 1. Translation Translation  Translation is a term used in geometry to describe a function that moves an object from one position to a new certain distance.  The object is not altered in any other way except translation.  Consider a point P (x, y) is translate to a new position P’(x’,v’) by a distance tx in X direction and ty in Y direction. Therefore, 𝑋 =𝑋+𝑡 𝑌 =𝑌+𝑡 The above equation can be written in matrix form as , 𝑋 𝑋 𝑡 = + 𝑡 𝑌 𝑌 2. Rotation Rotation is a term used in geometry to describe a function that rotates the object at particular angle 0 from its origin. Fig. shows that the point P(x, y) is located at angle from the horizontal X coordinate with distance ‘r’ from the origin. After rotating it at the angle θ with constant distance r we get new point P (x, y). Let, r = Constant distance of the point from the origin. ø = Original angular position of point from X coordinate. 3. Scaling Scaling of Rectangle 4. Reflection Reflection is transformation in which the mirror image can be obtained of original object. There are various types of reflection are as follows: 1) Reflection about Xaxis. 2) Reflection about Y-axis. 3) Reflection about origin. 4) Reflection about line y = x Reflection about X-axis 5. Shear Concatenated Transformation A Concatenated transformation is nothing but the combination of different geometric transformation. And it can be achieved by matrix multiplication of different geometric transformation. Advantage of using concatenated transformation is that we can combine the effects of two or more matrices by multiplying them. This means that, to rotate a model and then translate it to some location, we do not need to apply two matrices. Instead, you multiply the rotation and translation matrices to produce a composite matrix that contains all of their effects. This process, called concatenation transformation, Consider a point P(x, y) is to rotate at an angle 0 first at new coordinate P’(x’,y’) and then reflect about X- axis. Two-Dimensional Geometric Transformation using Homogeneous Coordinates In 2-D geometric transformation, translation can be done by matrix addition and other transformation such as rotation, scaling, shear, reflection can be done by matrix multiplication. So, any operation involving translation transformation in concatenated transformation is not possible. For making it possible, 2-D Homogeneous coordinate system represents coordinates in 2 dimensions with 3 vectors by adding third coordinate to every 2D point. This allows us to express all transformation equation as matrix multiplication, providing that we also expand the matrix representation for coordinate position. To express any 2-D transformation as a matrix multiplication, we represent each Cartesian coordinate position (x, y) with homogeneous coordinates (X h, Yh, w). Where, w is any non zero scalar factor. Homogeneous coordinate system can be applied to following 2-D geometric transformation. - Translation - Rotation - 3) Scaling - Reflection - Shear Two-Dimensional Inverse Transformation Two dimensional inverse transformation is very useful while performing different geometric transformation operation specially forming for concatenated matrix. Some of the two dimensional inverse transformation are as follows : 1. Inverse Translation 2. Inverse Rotating 3. Inverse Scaling 4. Inverse Reflection Coordinate System In CAD modeling, it is often required to transform the objects from one coordinate system to another. There are three major types of coordinate systems commonly used in CAD modeling. A. Model co-ordinate system (MCS) OR World co-ordinate system or Global co-ordinate system B. User co-ordinate system (UCS) or Local co-ordinate system or Working co-ordinate system (WCS) C. Screen co-ordinate system (SCS) A. Model Coordinate System (MCS) or World Coordinate System or Global Coordinate System : The CAD software stores the graphical information (coordinate data) in the model database with reference to the coordinate system known as model coordinate system or world coordinate system or global coordinate system. The model coordinate system or world coordinate system is the reference space of the model with respect to which all the geometrical data of model is stored in database. This is the only coordinate system that modeling software recognizes when storing or retrieving the graphical information in or from model database. B. User Coordinate System (UCS) or Local Coordinate System or Working Coordinate System (WCS): In CAD modeling, the user inputs the graphical information (coordinate data) with reference to ttic coordinate system known as user coordinate system Of local coordinate system or working coordinate system. If the geometric model has a complex geometry or I specific orientation, it is highly inconvenient ( to input the graphical data in a model coordinate system (MCU). C. Screen Coordinate System: Screen coordinate system is the 2D Cartesian coordinate system whose origin in located at the lower left corner of the graphics display screen, as shown in Fig. This system is the display-device dependent. Geometric Mapping The mapping which changes the graphical description of the graphic element from one coordinates system to another coordinate system without changing its relative position, orientation, size and shape. Two dimensional geometric mapping Types of 2D geometric mapping A. Translation mapping B. Rotational mapping C. General mapping A. Translational Mapping: Translational mapping : If the axes of the two coordinate systems (original and new) are parallel, then the mapping is called translational mapping B. Rotational mapping If the Iwo coordinate systems (original and new) have common origin but the axes arc at an angle, thcn (lie mapping is callcd rotational mapping C. General mapping It is the combination of rotational and translation mapping. Projection of geometric models The thrce-dimensional (3D) view of a geometric model needs to be displayed in two-dimensional (2D) form, as the display devices arc only two-dimensional (2D). Projection: It is a transformation used for transforming a three-dimensional (3D) model in to a two- dimensional form. Terminology used in projections: 1. Centre of projection: The centre of projection is a view point from where all the projection rays start. 2. Projectors: The projectors are the projection rays used for obtaining the projection of an object. 3. Projection plane: The projection plane is the plane on which the two-dimensional image of an object is obtained. Transformation for orthographic projection