A PRESENTATION ABOUT ENGINEERING CURVES WHICH INCLUDES BASIC CURVE DEFINITION AND DETAILS ABOUT HERMITE CURVES AND B SPLINES. DERIVATION , EQUATION ETC.

1. SARDAR VALLABHBHAI PATEL INSTITUTE
OF TECHNOLOGY
PREPARED BY : PANCHAL PARTH HITESHKUMAR (18MEMECG003)
GUIDED BY : Dr. P. H. SHAH

2. CURVES:
CURVES ARE DEFINED AS THE LOCUS OF A POINT MOVING WITH ONE
DEGREE OF FREEDOM.
CURVES ARE CATEGORIZED AS FOLLOWS :
a. BEZIER CURVES
b. HERMITE CURVES
c. B-SPLINE CURVES
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3. HERMITECURVES:
-ALGEBRAIC AND GEOMETRIC FORMS
- MATRIX FORM
- 3-POINT POINT INTERPOLATION
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4. ALGEBRAICANDGEOMETRICFORM:
THE ALGEBRAIC FORM IS GIVEN BY FOLLOWING THREE POLYNOMIALS:
x(u) = ax u3 + bx u2 + cx u + dx
y(u) = ay u3 + by u2 + cy u + dy ……………… (1)
z(u) = az u3 + bz u2 + cz u + dz
IN GENERAL VECTOR NOTATION OF EQUATION 1 IS :
p(u) = au3 + bu2 + cu + d ………….. (2)
WE USUALLY RESTRICT THE PARAMETER u TO VALUES IN THE INTERVAL 0 TO 1.
THIS RESTRICTION BOUNDS THE CURVE, CREATING A CURVE SEGMENT.
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THE GEOMETRIC FORM IS AS FOLLOWS:
p(u) = F1(u) p0 + F2(u) p1 + F3 (u) p0" + F4 (u) p1"
VECTORS p0, p1, p0“, p1“ ARE THE GEOMETRIC COEFFICIENTS.
‘F’ TERMS ARE THE HERMITE BASIS FUNCTIONS.

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MATRIXFORM :
p(u) = au3 + bu2 + cu + d
THE ABOVE EQUATION CAN BE WRITTEN AS FOLLOWS :
p(u) = [ u3 u2 u 1 ] [ a b c d ]T

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3 –POINTINTERPOLATION:
THERE IS A THREE POINT INTRPOLATION METHOD FOR DEFINING A CUBIC
HERMITE CURVE GIVEN THE FOLLOWING CONDITIONS:
THE TWO ENDPOINTS Po AND P1, AN INTERMEDIATE POINT Pi, AND ITS
CORRESPONDING BUT UNSPECIFIED PARMETRIC VARIABLE Ui AND UNIT
TANGENT VECTORS To AND Ti.
TO DETERMINE TANGENT VECTORS
Po” = KoTo and P1” = K1T1 , AND ATTEMPT TO FIND Ko AND K1 SUCH THAT THE
CURVE PASSES THROUGH P1

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WE HAVE ,

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B-SPLINES:
A B- spline curve differs from a Hermite curve in that it usually consist of more than one
segment. Each segment is defined and influenced by only A few control points.
The B-spline curve features local control and any desired continuity.
The B-spline curve is an approximating curve, so it is based on control points.

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QUADRATICUNIFORMB- SPLINE:

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REFERENCES:
1. GEOMETRIC MODELING
BY MICHAEL E. MORTENSON
2. COMPUTER GRAPHICS AND COMPUTER MODELING
BY DAVID SALOMON

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