Mathematical Modeling of Synthetic Curves

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MATHEMATICAL MODELING OF SYNTHETIC CURVES

IN MATLAB

Prepared by

Akshat Agha

2009A4PS380G

Prepared for

D.M. Kulkarni

In partial fulfillment of requirements of the course

BITS C331

Computer Oriented Project

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE

PILANI- K.K. BIRLA GOA CAMPUS

November, 2012


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ACKNOWLEDGEMENT

The report is written under the supervision Associate Professor Dr. D.M. Kulkarni, who has

given valuable inputs and suggestions. I also like to thank Harshit Pathak, Poorwa Shekhar,

Suraj R. Pawar, Krati Agarwal and Aditya Ramani who have worked on the same project.

Akshat Agha

2009A4PS380G


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ABSTRACT

The project is concerned with mathematical modeling of geometric curves using

MATLAB. Two categories of curves are modeled.

Analytic curves, which are conic sections, which were covered, are as follows:

1. Line2. Parabola3. Hyperbola4. Ellipse5. Circle

Synthetic curves, which include the category of splines, which were covered are as

follows:

1. Hermite Cubic Spline2. Bezier Curves

Each part of the report provides an introduction to the curve and its parametric form.

The code for the implementation of the curve on MATLAB is also included.


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INDEX

S.No Contents Page No.

1. Introduction 5

2. Geometric Modeling using MATLAB 6

2.1 Line 6

2.2 Circle 6

2.3 Parabola 7

2.4 Ellipse 82.5 Hyperbola 9

2.6 Synthetic Curves 10

2.6.1 Hermite Cubic Spline 10

2.6.2 Bezier Curve 11

3. References 13


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Network

CADManufacturing

INTRODUCTION

CAD (Computer Aided Design) is used all over the world by many different types of

engineering manufacturers. In industry, CAD refers to any computer software that isused to produce high quality drawings and models which meet exact specifications.

CAD software is often then linked to machinery to perform a task to manufacture part of

or a whole product; this is known as CAM (Computer Aided Manufacture).

Computer aided Design is a sub-process of Design process.

ComputerGraphics

GeometricModeling

DesignEngineering

CAD

CAM


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GEOMETRIC MODELING USING MATLAB

Geometric or surface modeling traditionally identifies a body of techniques that can model

certain classes of piecewise parametric surfaces, subject to particular conditions of shapeand smoothness. It developed as a separate field in several industries, including automobile,

aerospace, and ship building, and has some of its intellectual roots in approximation theory.

All CAD/CAM systems provide users with curve entities. Curve entities are divided into two

categories:

AnalyticPoints, lines, arcs, fillets, chamfers, and conics (ellipses, parabolas, and hyperbolas)

SyntheticIncludes various types of spline; Cubic spline, B-spline and Bezier curve

1. Line

A line connecting two end points P1 and P2 has a parametric equation:

P = P1 + u(P2 P1), where 0u1

global p1 p2 p3 num u y x l p q

p=[]

q=[]

num=inputdlg('Enter first point [x1 y1]')

p1=str2num(num{1})

num = inputdlg('Enter second point [x2 y2]')

p2=str2num(num{1})

u=0:0.1:1

y=p1(2) + u*(p2(2)-p1(2))

q=[q;y]

x=p1(1)+u*(p2(1)-p1(1))

p=[p;x]

plot(x,y)

%p3=p1 + u*(p2-p1)

%plot(u,p3)

axis equal

2. Circle

The basic parametric equation of a circle of radius r and center (xc,yc) is:

X = xc + r cos(u)

Y = yc + r sin(u), where 0u2


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global x y theta temp1 temp2 r u

temp1=inputdlg('Enter radius')

r=str2num(temp1{1})

temp2=inputdlg('Enter step value')

%'CIRCLE',1,'0.1', options)

u=str2num(temp2{1})theta=-pi:u:pi

x=r*cos(theta)

y=r*sin(theta)

plot(x,y)

axis equal

3. Parabola

Aparabola is the curve created when a plane intersects a right circular cone parallel to the

side (elements) of the cone.

The parametric equation in 2-D for a parabola is:

X = xv + Au2

Y = yv + 2Au, where 0u

global c a v

global x y ym u p q

x = []

y = []

prompt={'Vertex','Focal Length','Increment Value'};

name='Parabola';

numlines=1;

defaultanswer={'0 0','1','0.1'};

options.Resize='on';

options.WindowStyle='normal';

options.Interpreter='tex';

answer=inputdlg(prompt,name,numlines,defaultanswer,options);

c=str2num(answer{1});

a=str2num(answer{2});

v=str2num(answer{3});

u=0:v:10

p = u.*u

q = u.*a

x = c(1) + (a.*p)

y = c(2) + (2.*q)

ym=-y

hold on

plot(x,y)

plot(x,ym)axis equal


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4. EllipseAn ellipse is the curve created when a plane cuts all the elements (sides) of the cone but its

not perpendicular to the axis.

The basic equation of an ellipse with center (xc,yc) and radii a,b is:

X = xc + a cos(u)

Y = yc + b sin(u), where 0u2

%[output] = tablerot(v0,th0,mu,omg)

%DIALOGUE BOX GENERATION

prompt={'CENTER OF ELLIPSE X','CENTER OF ELLIPSE Y','MAJOR AXIS','MINOR AXIS'};

name='Simulation ELLIPSE';

%SETTING THE DEFAULT VALUES

numlines=1;defaultanswer={'0','0','1','1'};

options.Resize='on';options.WindowStyle='normal';


%INPUTTING THE VALUES


%CHANGING THE INPUTTED STRING INTO NUMERICAL VALUES

cx=str2num(answer{1});

cy=str2num(answer{2});


b=str2num(answer{4});N=1000;

TH=0.36;

for i=1:N

p(i)=cx+a/2*cos(i*TH);

q(i)=cx+b/2*sin(i*TH);

end

plot(p,q)

lm = max([max(abs(p)) max(abs(q))]);

lm = lm+2;

axis([-lm lm -lm lm]);

axis squarehold off


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5. Hyperbola

A hyperbola is the curve created when a plane parallel to the axis and perpendicular to the

base intersects a right circular cone.

The parametric equation in 2-D is:

X = xv + Acoshu

Y = yv + Bsinhu

prompt={'ABSICCA OF CENTER','ORDINATE OF CENTER','MAJOR AXIS','MINOR AXIS'};

name='Simulation HYPERBOLA';

numlines=1;defaultanswer={'0','0','1','1'};



options.Interpreter='tex'; answer=inputdlg(prompt,name,numlines,defaultanswer,options);

cx=str2num(answer{1});

cy=str2num(answer{2});


b=str2num(answer{4});

TH=0.1;

for i=-20:20

p(i+21)=cx+a*cosh(i*TH);

q(i+21)=cy+b*sinh(i*TH);

r(i+21)=cx-a*cosh(i*TH);

s(i+21)=cy-b*sinh(i*TH);

end

plot(p,q,r,s)

lm = max([max(abs(p)) max(abs(q))]);

lm = lm+2;

axis([-lm lm -lm lm]);

axis square


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6. Synthetic Curves

1. Hermite Cubic Spline

On the unit interval , given a starting pointp0 at and an ending point p1 atwith starting tangent m0 at and ending tangent m1 at , the polynomial

can be defined by :

where t [0, 1].

%Taking all relevant inputs from user

prompt={'First Point','Second Point','First Tangent','Second Tangent','Increment Value'};

name='Hermite Curve';

numlines=1;

defaultanswer={'3 2','11 5','5 0','5 0','0.1'};





p1=str2num(answer{1});

p2=str2num(answer{2});

t1=str2num(answer{3});

t2=str2num(answer{4});

inc=str2num(answer{5});

global h

%Defining Hermite matrix

h=[2 -2 1 1; -3 3 -2 -1; 0 0 1 0; 1 0 0 0]

global t p1 num p2 t1 t2 c inc

%Defining the working matrix of points and tangents

c=[p1;p2;t1;t2]

global x y s j temp1 m1 temp2 temp3

%Defining x and y as matrices of undefined size

x=[]

y=[]

%For loop for calculating values of x and y by incrementing s as per user input

for s =0: inc :1

t=[s^3 s^2 s 1]

temp1 = c(1:4,1:1)

m1 = t*h

temp3 = m1*temp1

x=[x;temp3]

temp2 = c(1:4,2:2)

temp3=m1*temp2

y=[y;temp3]

j=j+1

end

%Plotting all results to obtain final Hermite curve

plot(x,y);axis square


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2. Bezier Curve

Four points P0, P1, P2 and P3 in the plane or in higher-dimensional space define a cubic

Bzier curve. The curve starts at P0 going toward P1 and arrives at P3 coming from the

direction ofP2. Usually, it will not pass through P1 or P2; these points are only there to

provide directional information. The distance between P0 and P1 determines "how long" the

curve moves into direction P2before turning towards P3.

The parametric form of the curve is given as:

%DIALOGUE BOX GENERATION

prompt={'Number of points'};

name='Bezier curve';


numlines=1;defaultanswer={'4'};options.Resize='on';






n=str2num(answer{1});

count = 1;

px = [];

py = [];

for k=1:n

prompt={'x coordinate','y coordinate'};

name='Enter points';


numlines=1;defaultanswer={'1','1'};







px(count)= str2num(answer{1});

py(count)= str2num(answer{2});

count=count+1;

end


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ax=0;

ay=0;

count=1;

n=n-1;

for u=0:0.01:1

for i=0:n

b=(factorial(n)/factorial(n-i)/factorial(i))*(u^i)*(1-u)^(n-i);

ax=ax+px(i+1)*b;

ay=ay+py(i+1)*b;

end

qx(count)=ax;

qy(count)=ay;

ax=0;

ay=0;

count=count+1;

end

plot(qx,qy);


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References

1. www.mathworks.in2. www.math.utah.edu

Documents

Mathematical Modeling of Synthetic Curves