The document discusses concepts in computer-aided design and manufacturing, focusing on parametric curve representation, continuity conditions, and spline interpolation. It details different types of projections, including parallel and perspective transformations, and highlights the significance of polynomial fits in data approximation. The importance of continuity at curve junctions and the characteristics of interpolation versus approximation splines are also addressed.

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National Institute of Technology Jamshedpur
Department of Mechanical Engineering
Department of Mechanical Engineering
National Institute of Technology Jamshedpur
Deepak Kumar,
Department of Mechanical Engineering, NIT Jamshedpur
deepak.me@nitjsr.ac.in
Computer Aided Design/Computer Aided Manufacturing

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Department of Mechanical Engineering

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3D Scaling
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The position vector is assumed to be a row vector in right-handed system
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Polynomial Fit
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Parametric Continuity Conditions
• To represent a curve as a series of
piecewise parametric curves, these
curves to fit together reasonably …
Continuity!
National Institute of Technology Jamshedpur

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Continuity
When two curves are joined, we typically want some
degree of continuity across the boundary (the knot)
– C0
, “C-zero”, point-wise continuous, curves share
the same point where they join
Let C1(u) and C2(u) , be two parametric Curves.
0 1
u
 
C1(1) = C2(0)
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– C1
, “C-one”, continuous derivatives, curves
share the same parametric derivatives
where they join
C´1(1)= C´2(0)
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– C2
, “C-two”, continuous second derivatives,
curves share the same parametric second
derivatives where they join
– Higher orders possible
C˝1(1)= C˝2(0)
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Interpolation Splines
• When polynomial sections are fitted so that
the curve passes through each control point,
the resulting curve is said to interpolate the
set of control points.
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Approximation Splines
• When polynomial sections are fitted to the general
control point path without necessarily passing through
any control point, the resulting curve is said to
approximate the set if control points.

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Department of Mechanical Engineering

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Department of Mechanical Engineering

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Department of Mechanical Engineering

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PROJECTIONS
1. Parallel Projections
a) Orthographic Projections
b) Axonometric Projections
2. Perspective Transformations and
Projections
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Department of Mechanical Engineering

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PERSPECTIVE PROJECTIONS
1. Perspective Transformations and
Projections
a) Single point
b) Two point
c) Three Point
2. Vanishing points and trace points
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Department of Mechanical Engineering

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PLANE CURVES
1. Analytical Curves
2. Synthetic Curves
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Parametric Representation of Curves and Surfaces
Two types of equations for curve representation
(1) Parametric equation x, y, z coordinates are related by a parametric variable (uor θ)
(2) Nonparametric equation x, y, z coordinates are related by a function
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Curve Equations
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Department of Mechanical Engineering

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Parametric Equations –Advantages over nonparametric forms
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y
x
Curve Fitting
Data point approximated by straight Line
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Department of Mechanical Engineering

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Is a straight line suitable for each of these cases ?
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Department of Mechanical Engineering

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Polynomial Fit
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Department of Mechanical Engineering

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Department of Mechanical Engineering

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Department of Mechanical Engineering

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Department of Mechanical Engineering

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Parametric Continuity Conditions
• To represent a curve as a series of
piecewise parametric curves, these
curves to fit together reasonably …
Continuity!
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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Continuity
When two curves are joined, we typically want some
degree of continuity across the boundary (the knot)
– C0
, “C-zero”, point-wise continuous, curves share
the same point where they join
Let C1(u) and C2(u) , be two parametric Curves.
0 1
u
 
C1(1) = C2(0)
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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– C1
, “C-one”, continuous derivatives, curves
share the same parametric derivatives
where they join
C´1(1)= C´2(0)
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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– C2
, “C-two”, continuous second derivatives,
curves share the same parametric second
derivatives where they join
– Higher orders possible
C˝1(1)= C˝2(0)
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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Interpolation Splines
• When polynomial sections are fitted so that
the curve passes through each control point,
the resulting curve is said to interpolate the
set of control points.
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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Approximation Splines
• When polynomial sections are fitted to the general
control point path without necessarily passing through
any control point, the resulting curve is said to
approximate the set if control points.
National Institute of Technology Jamshedpur
Department of Mechanical Engineering