IV-1

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IV Projection

• A projection transformation moves from three
  dimensions to two dimensions
• Projections occur based on the viewpoint and the
  viewing direction
• Projections move objects onto a projection plane
• Projections are classified based on the direction
  of projection, the projection plane normal, the
  view direction, and the viewpoint
• Two primary classifications are parallel and
  perspective

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Parallel Projection

• All projectors run parallel and in the viewing
  direction
• Projecting onto the z 0 plane along the z axis
  results in all z coordinates being set to zero

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Orthographic Projection

• The simplest of the parallel projections, which
  is commonly used for engineering drawings
• They accurately show the correct or true size and
  shape of a single plane face of an object
• The matrix for projection
• onto the x0 plane
• The matrix for projection
• onto the y0 plane

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• The matrix for projection
• onto the z0 plane

• The centers of projection are at infinity
• A single orthographic projection does not provide
  sufficient information to visually and
  practically reconstruct the shape of an object

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Axonometric Projection

• An axonometric projection is constructed by
  manipulating the object, using rotations and
  translations, such that at least three adjacent
  faces are shown
• The center of projection is at infinity
• Unless a face is parallel to the plane of
  projection, an axonometric projection does not
  show its true shape
• The foreshortening factor is the ratio of the
  projected length of a line to its true length
• There are three axonometric projections of
  interest trimetric, diametric, and isometric

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• An isometric projection is a special case of a
  diametric projection, and a diametric projection
  is a special case of a trimetric projection
• Foreshortening factors along the projected
  principal axes are
• A trimetric projection is formed by arbitrary
  rotations, in arbitrary order, about any or all
  of the coordinate axes, followed by parallel
  projection onto the z0 plane

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• A dimetric projection is a trimetric projection
  with two of the three foreshortening factors
  equal the third is arbitrary
• For example, a dimetric projection is constructed
  by a rotation about the y-axis through an angle
  followed by rotation about the x-axis through an
  angle and projection from a center of
  projection at infinity onto the z0 plane

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Let
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• Isometric projection All three foreshortening
  factors are equal

• The foreshortening factor
• The angle that the projected x-axis makes with
  the horizontal is

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Oblique Projection

• An oblique projection is formed by parallel
  projectors from a center of projection at
  infinity that intersect the plane of projection
  at an oblique angle

• Only faces of the object parallel to the plane of
  projection are shown at their true size and shape

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• Cavalier projection The angle between the
  oblique projectors and the plane of projection is
  . The foreshortening factors for all three
  principal directions are equal to 1
• Cabinet projection The foreshortening factor for
  edges perpendicular to the plane of projection is
  one-half. The angle between the projectors and
  the plane of projection is
• The angle between the oblique projectors and the
  plane of projection is
• The transformation for
• an oblique projector is

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Perspective Projection

• The viewpoint is the center of projection for a
  perspective projection
• In perspective transformations parallel lines
  converge, object size is reduced with increasing
  distance from the center of projection

• All of these effects aid the depth perception of
  the human visual system, but the shape of the
  object is not preserved

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• Any of the first three elements of the fourth
  column of the general homogeneous
  coordinate transformation matrix is nonzero
• The single-point perspective transformation with
  centers of projection and vanishing points on the
  z-axis

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• The center of projection is
• Perspective factor r

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• Vanishing point

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• The single-point perspective transformation with
  centers of projection and vanishing points on the
  x-axis
• The center of projection is
• The vanishing point is

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• The single-point perspective transformation with
  centers of projection and vanishing points on the
  y-axis
• The center of projection is
• The vanishing point is

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• Two-point perspective transformation Two terms
  in the first three rows of the fourth column of
  the transformation matrix are nonzero.

• Two centers of projection
• Two vanishing points

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• Three-point perspective transformation Three
  terms in the first three rows of the fourth
  column of the transformation matrix are
  nonzero.

• Three centers of projection
• Three vanishing points

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• Consider simple translation of the object
  followed by a single-point perspective projection
  from a center of projection at onto the
  z0 plane

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• Combine rotation and translation about a single
  principal axis to obtain an adequate 3D
  representation

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• A three-point perspective transformation can be
  obtained by rotating about two or more of the
  principal axes and then performing a single-point
  perspective transformation
• For example, a rotation about the y-axis followed
  by a rotation about the x-axis and a perspective
  projection onto the z0 plane from a center of
  projection at zzc

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Vanishing Point

• Principal vanishing points Points on the
  horizontal reference line at which lines
  originally parallel to the untransformed
  principal axes converge, when a perspective view
  of an object is created by using a horizontal
  reference line, normally at eye level.
• In general, different sets of parallel lines have
  different principal vanishing points
• Trace points For planes of an object which are
  tilted relative to the untransformed principal
  axes, the vanishing points fall above or below
  the horizontal reference line. These are often
  called trace points.

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• The methods for determining the vanishing points
  
• The intersection point of a pair of transformed
  projected parallel lines

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• The transformation of the points at infinity on
  the principal axes

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• The transformation of the points at infinity for
  skew planes can be used to find trace point

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Reconstruction of 3D Images

• The general perspective transformation is

• The transformation projected onto the z0 plane
  is

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• If the location of several points which appear in
  the perspective projection are known in object
  space and in the perspective projection, then it
  is possible to determine the transformation
  elements

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• Project I Please show the effect of
  transformation, such as translation, rotation,
  reflection, parallel projection and perspective
  projection on a cube with one corner cut off.