CAD_transformation 3-dimensional presentation file

Transformation :3-dimesional

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National Institute of Technology Jamshedpur
Department of Mechanical EngineeringDepartment of Mechanical Engineering
National Institute of Technology Jamshedpur
Deepak Kumar,
Department of Mechanical Engineering, NIT Jamshedpur
[email protected]
Computer Aided Design/Computer Aided Manufacturing

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Department of Mechanical Engineering

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Department of Mechanical Engineering

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3D Scaling
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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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The position vector is assumed to be a row vector in right-handed system
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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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Department of Mechanical Engineering

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Department of Mechanical Engineering

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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)
–C
0
, “C-zero”, point-wise continuous, curves share
the same point where they join
Let C
1(u) and C
2(u) , be two parametric Curves. 01u
C
1(1) = C
2(0)
National Institute of Technology Jamshedpur

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

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–C
2
, “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

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Interpolation Splines
•When polynomial sections are fitted so that the
curvepassesthrough each control point, the
resulting curve is said to interpolatethe set
of control points.
National Institute of Technology Jamshedpur

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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
approximatethe 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.ParallelProjections
a)OrthographicProjections
b)AxonometricProjections
2.PerspectiveTransformationsand
Projections
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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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Department of Mechanical Engineering

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Department of Mechanical Engineering

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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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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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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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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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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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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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Department of Mechanical Engineering

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PLANE CURVES
1. Analytical Curves
2. Synthetic Curves
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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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Department of Mechanical Engineering

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Department of Mechanical Engineering

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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 (uorθ)
(2) Nonparametric equation x, y, z coordinates are related by a function
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Department of Mechanical Engineering

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Curve Equations
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Department of Mechanical Engineering

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Department of Mechanical Engineering

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Parametric Equations –Advantages over nonparametric forms
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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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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)
–C
0
, “C-zero”, point-wise continuous, curves share
the same point where they join
Let C
1(u) and C
2(u) , be two parametric Curves. 01u
C
1(1) = C
2(0)
National Institute of Technology Jamshedpur
Department of Mechanical Engineering

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–C
1
, “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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–C
2
, “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
curvepassesthrough each control point, the
resulting curve is said to interpolatethe 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
approximatethe set if control points.
National Institute of Technology Jamshedpur
Department of Mechanical Engineering