written by: itzik ben shabat technion - israel institute of technology faculty of mechanical...

Written by: Itzik Ben Shabat

Technion - Israel Institute of TechnologyFaculty of Mechanical Engineering

Laboratory for CAD & Lifecycle Engineering

Lab 6 : Transformations using Lab 6 : Transformations using MATLAB MATLAB


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




Straightforward Implementation

We wish to transform the blue square into the red rectangle




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





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





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;





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 MatrixS=[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?





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:





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?





Answer:

green – R*S*T

Yellow - S*R*T

(the scaling after rotation

creates the distortion)




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]) - TranslationM = makehgtform('scale',s) - Uniform scaleM = makehgtform('scale',[sx,sy,sz]) - nonuniform scaleM = makehgtform('xrotate',t) - rotate around x axis – t in radiansM = makehgtform('yrotate',t) - rotate around y axis – t in radiansM = makehgtform('zrotate',t) - rotate around z axis – t in radians




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 link





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





Your Final Drawing function should look something like this




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 tutorial




Sources and Refrences

Mathworks Documentation Center - http://www.mathworks.com/help/documentation-center.html



written by: itzik ben shabat technion - israel institute of technology faculty of mechanical...

Documents

cad lifecycle engineering

square vertices

matlab slide

square convert vertices

blue square

variables square

vertices sx

matrix s