Computer Aided Design

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Computer Aided Design

• GEOMETRIC TRANSFORMATIONS

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The topics of geometric transformations

• Transformations
• Translation
• Scaling
• Reflection
• Rotation
• Homogeneous Representation
• Concatenated Transformations
• Mapping
• Translational mapping
• Rotational mapping
• General mapping
• Mapping as changes in coordinate system
• Inverse transformation and inverse mapping
• Projections
• Orthographic projection
• Perspective projection
• Design and Engineering Applications
• Visualizations in kinematics, mechanisms,
  linkages and robotics

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TRANSFORMATION OF GEOMETRIC MODELS

• Transformation of geometric models implies a
  unique relationship between each point on the
  original model to one and only one point on the
  transformed model
• Pf(P, transformation parameters) ?unique
• Geometric transformation means the coordinate
  system is fixed and the model is moved in that
  same coordinate system from one location to the
  other location, by rigid-body motion
• It is possible to propose one single
  transformation matrix T that enables all the
  rigid-body transformations translation,
  scaling, reflection, rotation, clipping and
  windowing
• Such T should have certain specific properties
• It should be as general as possible
• Must apply to all rigid body transformations
  including clipping and windowing
• Both 2-D and 3-D transformations should be
  possible
• Homogeneous form is preferable, because only
  homogeneous form of transformation matrix can
  include translation in the general representation
  of PTP

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TRANSLATION

• Every entity of geometric model remains parallel
  to its initial position

Slopes and tangent vectors (both magnitude and
direction) remain same at all points on the
curve, only the coordinates of points change
d
P
P
X
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Show that the HCC is invariant in affine
translational transformation and also that
translational transformation does not change the
shape of HCC.
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Translation with Homogeneous Transformation matrix
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• Prove that a general curve such as a Bezier or
  B-Spline is to be translated, the invariance
  property is valid.
• Solution Invariance means translating the
  control points and then generating the curve give
  the same result as translating the original
  curve.
• Let d be the translation vector and Pi and Pi be
  the original and translated control points of the
  general curve.

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• Develop the translational transformation equation
  for
• (a) Hermite bicubic spline surface,
• (b) a bicubic Bezier surface,
• (c) bicubic B-spline surface

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SCALING

• Increase and decrease in size with/without change
  of shape is achieved by scaling. Scaling may be
  done w. r. t. origin of MCS or another point.
• Implemented as zoom operation

Tangent vectors get scaled by s
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Show that the scaling transformation does not
change the shape of HCC.
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REFLECTION
Reflection is useful - in building parts of
symmetry - only half the part needs to be
built - rest is built by reflection
Basically a negation process
Reflection may be done generally through(a)
either a plane(b) or a line(c) or a point
Particular cases of these are 1) Through one of
the principal planes (X0, Y0 or Z0) 2)
Through one of the principal axes (X, Y or Z) 3)
Through the origin of MCS
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Y
Y
X
X
Z
Z
Y
X
Z
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• Through a principal plane It is equivalent to
  negating the corresponding coordinate of each
  point on the entity.
• X0 plane gt m11 -1, m22m331
• Y0 plane gt m11m331 m22 -1
• Z0 plane gt m11m221 m33 -1
• 2) Through a principal axis equivalent to
  reflecting through the two principal planes
  intersecting at that axis
• X - axis gt m11 1, m22m33-1
• Y - axis gt m11m33-1 m22 1
• Z - axis gt m11m22-11 m33 1
• 3) Through the origin of MCS equivalent to
  reflecting through the three principal planes
  intersecting at the origin
• m11 m22m33 -1
• P - P
• Magnitudes of the tangent vectors remain same,
  directions are reversed.

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REFLECTION THROUGH A GENERAL PLANE
P
P1
Q
P2
P0
P
O
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REFLECTION THROUGH A GENERAL LINE
P
P1
Q
P0
P
O
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REFLECTION THROUGH A GENERAL POINT
P
Q
Pr
P
O
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Rotation

• Two aspects of rotation
• 1) Rotation of coordinate system x-y to x-y by
  positive ? and finding the coordinate values of
  the same point with respect to the new coordinate
  system.
• 2) Rotation of the point with respect to the same
  coordinate system by positive ?.
• The two are in opposite directions. Hence the
  minus sign for sine component exchanges.

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Y
P
X
Y
P
Y
P
?
a
X
X
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DIFFERENT PROBLEMS IN ROTATIONAL TRANSFORMATION

• Remember, rotation is always done about an axis,
  not a point. In the special case of planar
  rotation, the rotational axis appears to be a
  point.
• Different problems in rotational transformation
  are
• Rotation about each of the principal axes (of
  MCS)
• Two-dimensional rotation about an arbitrary axis
  that does not pass through the origin of MCS and
  is parallel to any one of the X, Y or Z principal
  axes
• Three dimensional rotation about an arbitrary
  axis that is inclined to all the principal axes
  of MCS and passes through the origin of MCS
• The most general case of 3-D rotation about an
  arbitrary axis that is inclined to all the three
  principal axes of MCS and does NOT pass through
  the origin of MCS

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• Rotation about each of the principal axes (of MCS)

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2) Two-dimensional rotation about an arbitrary
axis that does not pass through the origin of MCS
and is parallel to any one of the X, Y or Z
principal axes
P
Y
Y
P1
P
?
P
P1
-P1
X
X
Method 1
Y
Pt
Pt
?
X
O, P1t
P
Y
Pt
P1
X
O, P1t
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3) Three dimensional rotation about an arbitrary
axis that is inclined to all the principal axes
of MCS and passes through the origin of MCS
Y
P
n
P
R
?
R
Q
r
s
j
i
X
k
Z
P
P
R
?
R
Q
r
m
s
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Special Characteristics of R(1) It is
skew-symmetric. (2) Its determinant is equal to
one. In general, the determinant of any
rotational transformation matrix describing
rotation about an axis passing through the origin
is equal to one. (3) Rx, Ry and Rz are
special cases of R. (how do you prove?
Substitute ni, nj, and nk, respectively, in
R)
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(4) The most general case of 3-D rotation about
an arbitrary axis that is inclined to all the
three principal axes of MCS and does not pass
through the origin of MCS
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• INVARIANCE UNDER ROTATIONAL TRANSFORMATION OF A
  POINT LYING ON THE AXIS OF ROTATION Any point
  lying on the axis of rotation itself, does
  remains itself after rotation.
• This point has a lot of practical importance.
• When a model is, for example, rotated about an
  axis which is coincident with one of its edges,
  then all the points lying on that edge of the
  model remain themselves after the rotation.
• CAD/CAM packages allow VERIFY ENTITY feature that
  can be used to check that the end points or any
  point on the edge remain itself after rotation.
• If you observe any small differences in
  insignificant decimal points, they are due to
  numerical errors. This is because processing of
  rotational transformation is done on the end
  points of that edge also.
• How do you prove?

Y
n
P
A
?
A
P
X
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REPRESENTATION USING HOMOGENEOUS COORDINATES

• It is essential to be able to represent all
  geometric transformations as matrix
  multiplications rather than additions or
  subtractions for concatenating different
  transformations that are done in sequence.
• As we have seen, translation does not readily
  does not allow it.
• Representing points in homogeneous coordinates is
  the effective answer to this problem.
• Homogeneous coordinates have enabled geometric
  transformations to be embedded in the graphics
  processor/accelerator (hardware) thus enormously
  speeding up the execution of display commands.
• Homogeneous coordinates are also useful in
  obtaining perspective views of geometric models,
  projective geometry, mechanism analysis and
  design and robotics.
• Homogeneous coordinates remove many anomalous
  situations such as representing points at
  infinity and the non-intersection of parallel
  lines, otherwise encountered with simple
  Cartesian coordinates.
• Rational parametric curves and surfaces can be
  more simply represented using homogeneous
  coordinates.
• An n-dimensional space is mapped into
  (n1)-dimensional space where the (n1)th
  dimension being the scaling factor h.
• For the purposes of geometric modeling, h1 is
  taken to avoid unnecessary divisions.

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CONCATENATED TRANSFORMATIONS
GENERALIZED 3-D ROTATION MATRIX
Is the concatenated R equivalent to the
generalized 3-D rotation matrix derived earlier?
Yes Example 9.5. What is the difference between
them? Concatenation is possible for all
transformations.
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Mapping of Geometric Models

• Mapping is equivalent to the reverse rotation we
  discussed earlier.
• The object or each of its points remain fixed in
  space but its description is changed to another
  coordinate system
• It is useful in many applications including
  finite element modeling
• Pf (P, mapping parameters)

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Translational mapping

• If the vector by which translation is done is d,
  then the same homogeneous translational
  transformation vector derived earlier will hold.

P
P
d
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Rotational Mapping

• It is the same as the reverse rotational
  transformation we discussed earlier

P
?
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General mapping

• The generalized mapping matrix is same as the
  generalized transformation matrix except we
  substitute the appropriate values in the
  elemental position.

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Mapping as Change of Coordinate System

• Changing the coordinate system so that the
  coordinates of the points in the transformed set
  with respect to the new coordinate system (XYZ)
  are the same as the coordinates of the points in
  the original set with respect to the original
  system (XYZ).
• The actual problem is however to determine the
  coordinates of such mapped points w. r. t. the
  XYZ.
• Applications Model merging or building solid
  models

Z
Y
Y
P
X
P
X
Z
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Z
Y
Y
P
X
P
X
Z
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Inverse Transformations and mappings

• Inverse transformations are important using which
  the the CAD/CAM system provides users with
  functions that can take an existing entity and
  return its coordinates relative to a given WCS.
• Normally the verify entity command in CAD
  packages returns the coordinates relative to the
  MCS

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Determining n and ? if R is given
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Types of Plane Geometric Projection

• Parallel Projection (C is at infinity)
• Orthographic Projection
• Axonometric Projection
• Trimetric
• Dimetric
• Isometric
• Oblique Projection
• Cavalier
• Cabinet
• Perspective Projection (C is at a finite
  distance)
• Single Point
• Two Point
• Three Point

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PROJECTIONS OF GEOMETRIC MODELS

• Two entities define projection
• A center of projection
• A plane of projection
• Two types of projection exist
• Parallel Projection
• Orthographic projection
• Direction of projection is normal to the
  projection plane
• Projection planes can be the principal planes or
  planes other than those
• Oblique projection
• Direction of projection is inclined to the
  projection plane
• Perspective Projection

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Parallel projection

• Parallel projection
• Center of projection is at infinity
• All projectors (projecting rays or lines) are
  parallel
• Parallel lines in the object remain parallel
  (preserves parallelism)
• Preserves actual dimensions and shapes of objects
• Angles are preserved only those faces of the
  object that are perpendicular to the projection
  plane

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Perspective Projection

• Perspective Projection
• Creates an artistic effect that adds some realism
  to perspective views
• The size of an entity is inversely proportional
  to its distance from the center of projection
  the closer the entity to the center, the larger
  its size is.
• This projection is not popular with engineers and
  draftsmen because actual dimensions and angles of
  objects and therefore shapes cannot be preserved.
• Therefore, measurements cannot be taken from
  perspective views directly.
• This projection cannot preserve parallelism
• More polular in architecture and fine arts

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The viewing coordinate system
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Orthographic Projection
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General Methodology of Orthographic Projection

• The viewing coordinate system (VCS) is fixed and
  the object and its MCS are rotated appropriately
• The rotation is done such that the required view
  plane of the object inits MCS becomes coincident
  with the xv-yv plane of the VCS
• Thus, for front view, there is no rotation
  required whereas the other two, top and side
  views require rotation

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Front View

• For front view, the XY and XvYv planes are
  already identical.
• To obtain this view, we simply project the
  geometry onto the viewing plane

• Pv is the point expressed in the VCS
• For front view, xvx and yvy

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Top view

• For top view, we need to rotate the model about
  the Xv axis by 900 degrees so that the XZ plane
  coincides with the Xv-Yv plane.
• This is followed by setting the y coordintes of
  the resulting points equal to zero.
• Why set y coordinates to be equal to zero?
  Because the Y axis of the MCS coincides with the
  projection direction.
• xvx and yv-z

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Right Side View

• The right side view is obtained by rotating the
  model and its MCS about Yv axis by 900 and
  setting the x coordinate to be zero.
• The viewing coordinates xv-z and yvy, and the
  transformation matrix for the right side view
  becomes the following.

• The T in all views is a singular matrix a
  column being completely zero.

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Axonometric Projection
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Isometric View (projection)

• The model and its MCS are customarily rotted at
  angle 45 or -45 degrees about the Yv axis
  followed by a rotation of 35.26 or -35.26
  degrees about the Xv axis.
• In practice, instead, 30 degrees is used instead
  of 35.26 degrees to simplify the job of the
  draftsman.

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Perspective Projection

• We place the center of projection, C, along the
  Zv axis at a distance of d from the origin of VCS
  and project onto the zv0 plane.
• We introduce the eye coordinate system (ECS).
• The transformation matrix of coordinates of
  points from the VCS to ECS or vice versa can be
  written as follows.

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