1. Written by: Itzik Ben Shabat Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering Lab 6 : Transformations using MATLAB

2. Written by: Itzik Ben Shabat Implementation of Dry Exercise Using MATLAB Transform a Square with parameters width =1 Height=1 Center at (0,0) Into a rectangle with the following parameters Width=0.5 Height=2 Center at (2,2) Rotated around Z axis by 45 deg Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

3. Written by: Itzik Ben Shabat Straightforward Implementation We wish to transform the blue square into the red rectangle Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

4. Written by: Itzik Ben Shabat Implementation of Dry Exercise Solution outline Calculate square vertices and initialize parameters Plot square Convert vertices to homogeneous coordinates Matrix multiplication (notice ordering) plot rectangle Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

5. Written by: Itzik Ben Shabat Straightforward Implementation Open a new.m file and Create a function called main6 which opens a new figure and axis in it and calls the drawingCB function [ ] = main6( ) %main6 Demonstrates the use of Basic Transformations using matrix %multiplications clear all; close all; clc; %code %Open figure and create axis Figureh=figure('NumberTitle','off','Name','Transformation Example',... 'Position',[200 200 500 500]); %bg is set to red so we know that we can only see the axes Axesh=axes('XLim',[-3 3],'YLim',[-3,3]); DrawingCB( ); end Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

6. Written by: Itzik Ben Shabat Straightforward Implementation Create a function called drawingCB which performs as the solution outline demonstrated Calculate square vertices and initialize parameters %Initializing Variables square=[-0.5 -0.5;-0.5 0.5;0.5 0.5;0.5 -0.5]; %represented by its vertices Sx=0.5; Sy=2; Tx=2; Ty=2; teta=45; Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

7. Written by: Itzik Ben Shabat Straightforward Implementation Plot square line([square(:,1);square(1,1)],[square(:,2);square(1,2)],'Color','b','LineWidth',3); Convert to homogeneous coordinates Hsquare=[square, ones(size(square,1),1)]; Generate Matrix S=[Sx 0 0;0 Sy 0 ;0 0 1]; T=[1 0 Tx;0 1 Ty; 0 0 1]; R=[cos(teta) -sin(teta) 0;sin(teta) cos(teta) 0; 0 0 1]; Question: What is the right matrix order of multiplication? Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

8. Written by: Itzik Ben Shabat Straightforward Implementation Answer: scale first, then rotate, then translate. %Calculate rectangle vertices Hrect=T*R*S*Hsquare'; Hrect=Hrect'; line([Hrect(:,1);Hrect(1,1)],[Hrect(:,2);Hrect(1,2)],'Color','r','LineWidth',3); grid on; Result: Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

9. Written by: Itzik Ben Shabat Straightforward Implementation Lets explore what are the results if we used a different order: Question: Can you guess which order is represented in green and which in yellow ? Hint : what is special about the yellow? Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

10. Written by: Itzik Ben Shabat Straightforward Implementation Answer: green – R*S*T Yellow - S*R*T (the scaling after rotation creates the distortion) Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

11. Written by: Itzik Ben Shabat MATALB Implementation – The right way In previous section we explored the implementation of transformations using matrix multiplication In MATLAB there are built in routines that make our lives easier ( no need to define the entire matrix and multiply it or transform to homogeneous coordinates) Makehgtform Create transform matrices for translation, scaling, and rotation of graphics objects M = makehgtform('translate',[tx ty tz]) - Translation M = makehgtform('scale',s) - Uniform scale M = makehgtform('scale',[sx,sy,sz]) - nonuniform scale M = makehgtform('xrotate',t) - rotate around x axis – t in radians M = makehgtform('yrotate',t) - rotate around y axis – t in radians M = makehgtform('zrotate',t) - rotate around z axis – t in radians Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

12. Written by: Itzik Ben Shabat MATALAB Transformation implementation h = hgtransform('PropertyName',propertyvalue,...) We will mainly use h = hgtransform(‘Parent’,ax,‘matrix’,M); Executes the transformation defined by the matrix M on all children (graphic objects) of ax. Ax can be an axes, hggroup or hgtransform objects For more information on hgtransform properties visit this linklink Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

13. Written by: Itzik Ben Shabat MATALAB Transformation implementation Implement in previous section example Guidance: Plot 2 squares and save their handles Calculate matrix Set hgtform object to parent one of the squares Refresh the figure using drawnow Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

14. Written by: Itzik Ben Shabat MATALAB Transformation implementation Your Final Drawing function should look something like this Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

15. Written by: Itzik Ben Shabat Final notes We recommend that you use this code to experiment and further investigate the nature of transformations Do not use the straightforward implementation for standard transformation in future codes (it is only an example to help you understand the principles) It is important to understand by now the object handles principles (figure,axis, graphic objects, groups etc.) It is important to understand child-parent relations between objects (visit link) – more on this in hierarchy tutoriallink Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering

16. Written by: Itzik Ben Shabat Sources and Refrences Mathworks Documentation Center - http://www.mathworks.com/help/documentation-center.html http://www.mathworks.com/help/documentation-center.html Technion - Israel Institute of Technology Faculty of Mechanical Engineering Laboratory for CAD & Lifecycle Engineering