technion - israel institute of technology faculty of mechanical engineering laboratory for cad &...

Technion - Israel Institute of TechnologyFaculty of Mechanical Engineering

Laboratory for CAD & Lifecycle Engineering

Lab 9: 3D & ProjectionsLab 9: 3D & ProjectionsBasicsBasics



Tutorial contents

Introduction to ProjectionsBasic ConceptsProjection Types and Matrix

OrthographicPerspective

Solve Examples



Introduction to Projections

We live in a 3D world but the screen where the rendered objects appear is a 2D space. Thus, we need to convert somehow our 3D world to 2D.

The operation which does this conversion is called projection.

When Programming we focus on 2 main types of projections Orthographic – casts parallel rays of light from the scene to projection plane Perspective – casts angled rays of light from the scene to projection plane

Perspective projection generates a realistic effect since our vision works the same way



Basic Concepts

camera coordinatesystem

viewport coordinatesystem

device/screencoordinate system



Projection Type - Orthographic

r0 – a vector pointing from world origin to Projection plane origin

r – object point vector (from world origin to P)

P – object point P’ – projected point U – view direction Z’ – distance from P to P’ along U r’ – P’ coordinates in word CS u1,u2 – projected image main axis

x’,y’ – P’ coordinate in image CS

P P’



Projection Matrix - Orthographic

When U is Orthogonal to u1 and u2 we can simplify the equations to get the matrix

If U is not orthogonal (like in oblique parallel) we must use the equations given in the lecture

1

2

0

0

0

0 0 0 1 0 0 0 1

x

yOrthographic

z

u r

u I rM

u r

0 2 0 1 0 1 2

1 2 2 1 1 2

( )( ) ( )( ) ( )( )' , ' , '

( ) ( ) ( )

r r u u r r u u r r u uX Y Z

u u u u u u u u u



Projection Type - Perspective

rv – eye coordinates

Other vectors defined above



Projection Matrix

By selecting rv=r0+du

We can use the matrix form

We can see that d causes a scaling effect

1

2

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0 0 0 1 0 0 0 1

x

yPerspective

z

d u r

d u I rM

d u r

d



Solve Examples

Given a cube in 3D spacex = [0 1 1 0 0 0 1 1 0 0 1 1 1 1 0 0]-0.5;

y = [0 0 1 1 0 0 0 1 1 0 0 0 1 1 1 1]-0.5;

z = [0 0 0 0 0 1 1 1 1 1 1 0 0 1 1 0]-0.5;

We wish to find the isometric projection so we can see the cube in a 2D plane



Solve Examples

From the lectures – Isometric projection

The rotation around y axis is φ=45 The rotation around x axis is θ=35.2



Solve Examples

Plot the 3D space for visualization

%plot cube in 3D space

Figureh=figure('NumberTitle','off','Name','3D space',...

'Position',[200 300 500 500]);

line(x,y,z);

view(3);

daspect([1,1,1]);

set(gca,'XLabel',text('string','X'),'YLabel',text('string','Y'),'ZLabel',text('string','Z'));



Solve Examples

Task:

Define the projection matrix



Solve Examples

Solution:

Set the projection matrix

MIsometric=[cosd(phi) 0 sind(phi) 0; sind(teta)*sind(phi) cosd(teta) -sind(teta)*cosd(phi) 0;...

-sind(phi)*cosd(teta) sind(teta) cosd(teta)*cosd(phi) 0;0 0 0 1];

Task

Convert to homogeneous coordinates[m,n] = size(x);

x4d = [x(:),y(:),z(:),ones(m*n,1)]'; %homographic coordinates



Solve Examples

Solution

Convert to homogeneous coordinates[m,n] = size(x);

x4d = [x(:),y(:),z(:),ones(m*n,1)]'; %homographic coordinates



Solve Examples

Task

Calculate the points location in 2D and plot



Solve Examples

Solution

Calculate the points location in 2D and plot

%project to 2D

x2d = MIsometric*x4d;

x2 = zeros(m,n); y2 = zeros(m,n);

x2(:) = x2d(1,:);

y2(:) = x2d(2,:);

Figureh=figure('NumberTitle','off','Name','2D space',...

'Position',[800 300 500 500]);

plot(x2,y2)

set(gca,'XLim',[-1 1],'YLim',[-1 1],'XLabel',text('string','u1'),'YLabel',text('string','u2'));



Solve Examples

Output – Question : Why is there a difference?



Solve Examples

Answer: which view is MATLAB default? Probably not isometric



Notes and References

This is not (!!!) the way to use projections in MATLAB This is just an illustrated example of how projections work.

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

technion - israel institute of technology faculty of mechanical engineering laboratory for cad &...

Documents