G43 – Geometric Modeling

Mathematical modeling of

Curves – Hermite Cubic Spline

Curves

1

Dr.C. ParamasivamE-mail: cpmech@tce.edu

Curves

1. Define the coordinate system for the development

of models based on input and geometry.

2. Develop and manipulate the curves and surfaces

using parametric equations.

3. Develop and manipulate the solid models using

Course Objectives

By the end of the course, student will be able to:

3. Develop and manipulate the solid models using

modeling techniques.

4. Implement the transformation and projection over

the geometric models.

5. Implement the neutral file formats over 2D

wireframe models.

Mathematical modeling of

Curves – Hermite Cubic Spline

Curves

By the end of the course, studentwill be able to:

Develop and manipulate thecurves and surfaces usingparametric equations.

Types of curve equations

1. Parametric equation:

Example: circle equation:

X = Rcosθ Y = Rsin θ z = 0 0 ≤ θ ≤ 2π

(the coordinates are defined with the help of the extra

parameter θ).

2. Non-parametric equation

1. Implicit form: x2+y2-R2=0 z=0.

2. Explicit form y=±√(R2-x2) z=0.

( these equations defined the x , y and z coordinates without the

assistance of extra parameters)

The parametric equation is the most popularform for representing curves and surfaces inCAD systems.

WIREFRAME MODELS

A wireframe model of an object is the

simplest, but more verbose, geometric model

that can be used to represent it

mathematically.

Curve Modeling

mathematically.

A wireframe

representation is a

3D line drawing of

an object

1. Analytical curvesThe curves which are having rigid form of equation without any flexibility to modify its original shapes after display.

Example:

1. Point

Types of curves

1. Point

2. Line, Line segment

3. Conic sections – Circle, Ellipse, parabola, hyperbola.

4. Fillet

5. Chamfer

2. Synthetic curves• The curves which are usually described by a

polynomial equation having flexibility to modify itsoriginal shape after display.

• A parameter (u) is used to control its shape anddegrees-of-curves.

Examples:

1. Hermite cubic spline curve1. Hermite cubic spline curve

2. Bezier curve

3. B-spline curve

4. Rational curve

a) Rational Hermite cubic spline

b) Rational Bezier curve

c) Rational B-spline curve

Synthetic Curves

• Analytic curves are usually not sufficient to meetgeometric design requirements of mechanical parts.

Example: Car bodies, ship hulls, airplane wings,propeller blades, shoe insoles, and bottles etc.

Need for synthetic curves

• When a curve is represented by a collection ofmeasured data points.

• When an existing curve must change to meetnew design requirements.

• When an existing curve must change to meetnew design requirements.

1. Hermite cubic spline curve

• Parametric spline curves are defined aspiecewise polynomial curves with certainorder of continuity.

• The parametric cubic spline curve connects• The parametric cubic spline curve connectstwo data (end) points and utilizes a cubicequation.

General condition required:

1. Two end points

2. Two end slopes

• The parametric equation of a cubic splinesegment is given by:

where, u – parameter

Ci - Polynomial coefficients

The scalar form of equation:

10)(3

0

≤≤∑==

uuCuPi

ii

The scalar form of equation:

X(u) = C3x u3 + C2x u2 + C1x u + C0x

y(u) = C3y u3 + C2y u2 + C1y u + C0y

z(u) = C3z u3 + C2z u2 + C1z u + C0z

The vector form of equation:

P(u) = C3u3 + C2u2 + C1u + C0

P0 – Starting point

P1 – End point

P’0 – Starting slope

P’1 – Starting slope

The matrix form of equation:

P(u) = UTC

where,

U = [u3 u2 u1 1]T

C = [C3 C2 C1 C0]T (coefficient vector)

Tangent Vector

10)('1

3

0

≤≤∑= −

=

uuiCuPi

ii

To find the coefficients Ci

Apply the known boundary conditions

P0 , P’0 at u = 0

P1 , P’1 at u = 1

Position and slope equation are:Position and slope equation are:

P(u) = C3u3 + C2u2 + C1u + C0

P’(u) = 3C3u2 + 2C2u + C1

When applying u = 0 and u = 1 on the position

and slope equations:

• P0 = C0

• P’0 = C1

• P1 = C3 + C2 + C1 + C0• P1 = C3 + C2 + C1 + C0

• P’1 = 3C3 + 2C2 + C1

After solving the above four equations bysimultaneous solution method the coefficientsare:

C0 = P0

C1 = P’0

C2 = 3(P1-P0) -2(P’0 - P’

1)C2 = 3(P1-P0) -2(P 0 - P 1)

C3 = 2(P0-P1) + P’0 + P’

1

After substituting all four coefficient, the

final blending functions are:

P(u) = (2u3 – 3u2 + 1)P0+ (-2u3 + 3u2) P1

+ (u3 – 2u2 + u)P’0 + (u3 – u2) P’

1

P’(u) = (6u3 – 6u)P0+ (-6u2 + 6u) P1 +P’(u) = (6u – 6u)P0+ (-6u + 6u) P1 +

(3u2 – 4u + 1)P’0 + (3u2 – 2u) P’

1

P(u) = UTC

P(u) = UT [MH]V

Where,

[MH] = Hermite matrix

V= Geometry (Boundary) vector

V= [P0

P1

P’0

P’1

]T

−−−

−

=

0001

0100

1233

1122

][H

M

Comparing the above two equations:

UTC = UT [MH]V

C = [MH]V

Final position and slope equation in the Final position and slope equation in the

matrix are:

P(u) = UT [MH]V

P’(u) = UT[MH]uV

Modification of resultant curve shape

1.By changing control point(s)

2.By changing end slope(s)2.By changing end slope(s)

1.By changing control point(s)

2. By changing end slope(s)

Hermite Cubic Spline curve using AutoCAD software

Characteristics of Hermite cubic spline

curve

1. The resultant shape will pass through all the

given data or control points.

2. It uses interpolation technique for curve

generation.generation.

3. The resultant curve has tangential property

with start and end slopes.

4. It has only global control.

5. The resultant curve has always cubic curve.

Limitations

1. It has only global control (or) Lack of local.

2. Always it has cubic degree.

3. It is not possible to apply, when higher

degree of curves required.degree of curves required.

Answer

Sample questions

1. Find the shape of the hermite cubic spline curve usingP0(10, 20) and P1(30, 50) with an inclination of 300 incounter clockwise direction? Plot your result on the graphsheet.

2. Plot the resultant shape of hermite cubic spline curve usingP0(0, 20) and P1(50, 50) with 300 inclination at the startingP0(0, 20) and P1(50, 50) with 30 inclination at the startingpoint and -450 inclination at the end point.

3. Derive the blending function of hermite cubic spline curve.

4. Describe the important properties of hermite cubic splinecurve.

5. Illustrate the method of modifying the resultant hermitecurve shape.