Tilting is defined as a rotation about the xaxis followed by rotation about the

Question

Tilting is defined as a rotation about the xaxis followed by rotation about the y-axis. A pentagon is subjected to a tilt of 30o about the x-axis and 20o about the y-axis. The 3-D pentagon has 10 vertices in (x, y, z): A(8,0,2), B(2,0,2), C(-1,5,2), D(3,9,2), E(8,6,2), F(8,0,-3), G(2,0,-3), H(-1,5,-3), I(3,9,-3), J(8,6,-3). Determine the final transformed coordinates of the pyramid after tilting: A’, B’, C’, etc. Also, determine the total transformation matrix [TT] which transforms the original vertices directly to the final verticesit is a single matrix. Caution: use radians if you use “sin” or “cos” commands in MATLAB or you can use “sind” or “cosd” for arguments in degrees.

Explanation / Answer

Find the required MATLAB code as the following:

%================================================

clear all;
clc;
alpha=20;
beta=30;
A=[8,0,2];
B=[2,0,2];
C=[-1,5,2];
D=[3,9,2];
E=[8,6,2];
F=[8,0,-3];
G=[2,0,-3];
H=[-1,5,-3];
I=[3,9,-3];
J=[8,6,-3];
cord = [A;B;C;D;E;F;G;H;I;J]';
% Rotation about X axis
rx = [1 0 0;0 cosd(alpha) -sind(alpha);0 sind(alpha) cosd(alpha)];

% Rotation about Y axis
ry = [cosd(beta) 0 sind(beta);0 1 0;-sind(beta) 10 cosd(beta)];

% Calculating titled coordinates
cord_tilt=ry*(rx*cord);
cord_tilt=cord_tilt';
display('The titled coordiantes are:');
display(cord_tilt);
% Calculating total transformation matrix
TT=(ry*rx);
display('The total transformation matrix is:');
display(TT);

%================================================

Sample output:

The titled coordiantes are:

cord_tilt =

    7.8679   -0.6840   -9.2128
    2.6717   -0.6840   -6.2128
    0.9287    4.0144   43.7528
    5.0769    7.7732   80.5253
    8.8940    4.9541   48.9459
    5.5187    1.0261    3.8192
    0.3225    1.0261    6.8192
   -1.4205    5.7245   56.7848
    2.7276    9.4833   93.5573
    6.5447    6.6642   61.9780

The total transformation matrix is:

TT =

    0.8660    0.1710    0.4698
         0        0.9397   -0.3420
   -0.5000    9.6931   -2.6064

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