Fundamentals of Orthographic Projections; Projections of lines
Lect24 Projections 1
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Transcript of Lect24 Projections 1
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159.235 Graphics 1
159.235 Graphics & Graphical
Programming
Lecture 24 - Projections - Part 1
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Projections - Outline
3D Viewing
Coordinate System & Transform Process
Generalised Projections
Taxonomy of Projections
Perspective Projections
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3D Viewing
Inherently more complex than 2D case.Extra dimension to deal with
Most display devices are only 2D
Need to use aprojection to transform 3Dobject or scene to 2D display device.
Need to clip against a 3D view volume.
Six planes.View volume probably truncated pyramid
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Coordinate Systems & Transform
ProcessObject coordinate systems.
World coordinates.
View Volume
Screen coordinates.
Raster
Transform
Project
Clip
Rasterize
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Generalised Projections.
Transforms points in a coordinate system of dimension n
into points in one of less than n (ie 3D to 2D)
The projection is defined by straight lines calledprojectors.
Projectors emanate from a centre of projection,pass
through every point in the object and intersect a
projection surface to form the 2D projection.
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Projections. In graphics we are generally only interested in planar
projectionswhere the projection surface is a plane.
Most cameras employ a planarfilm plane.
But the retina is not a plane - future devices such asdirect retina devices may need more complex projections
We will only deal with geometric projectionsthe
projectors are straight lines.
Many projections used in cartography are either non-
geometric or non-planar.
ExceptionImage-based rendering - advanced topic
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Projections.
Henceforth refer to planar geometric projections as just:
projections.
Two classes of projections :
Perspective. Parallel.
A
B
A
B
A
B
A
B
Centre ofProjection.
Centre of
Projection
at infinity
Parallel
Perspective
Parallel
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A Taxonomy of Projections
Planar geometric projections.
Parallel Perspective
Orthographic Oblique 1 point
2 point
3 point
Axonometric
Isometric
CavalierCabinet
Elevations
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Perspective Projections. Defined by projection plane and centre of projection.
Visual effect is termedperspective foreshortening.
The size of the projection of an object varies inversely withdistance from the centre of projection.
Similar to a camera - Looks realistic !
Not useful for metric information
Parallel lines do not in general project as parallel.
Angles only preserved on faces parallel to the projectionplane.
Distances not preserved
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Perspective
The first ever painting(Trini ty w ith the Virgin ,
St. John and Donors)
done in perspective by
Masaccio, in 1427.
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Perspective Projections
A set of lines not parallel to
the projection plane
converge at a vanishing
point.
Can be thought of in 3D as theprojection of a point at
infinity.
Homogeneous coordinate is 0
(x,y,0)
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Perspective Projections
z
x
y
Projection plane
xz
y
Lines parallel to a principal axis converge at an axisvanishing point. Categorized according to the number of such points
Corresponds to the number of axes cut by the projection plane.
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1-Point Projection
Projection plane cuts 1
axis only.
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1-Point Perspective
A painting (The
Piazza of St. Mark,
Venice) done byCanaletto in 1735-
45 in one-point
perspective
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2-Point Perspective
y
z x
Projection plane
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2-Point Perspective
Painting in two point
perspective by
Edward Hopper
The Mansard Roo f
1923 (240 Kb);
Watercolor on paper,13 3/4 x 19 inches;
The Brooklyn
Museum, New York
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3-Point PerspectiveGenerally held to add little beyond 2-point perspective.
y
z x
Projection plane
A painting (City
Night, 1926) by
Georgia O'Keefe, that
is approximately in
three-point
perspective.
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Intro to Projections -Summary 3D Viewing
Coordinate System & Transform Process
Generalised Projections
Taxonomy of Projections
Perspective Projections Clipping can be done in image
space if more efficientapplication dependent.
Parallel Projections next Acknowledgement - Thanks to Eric McKenzie, Edinburgh, from whose Graphics
Course some of these slides were adapted.
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Parallel Projections
Specified by a direction to the centre of projection,rather than a point.
Centre of projection at infinity.
Orthographic
The normal to the projection plane is the same as thedirection to the centre of projection.
Oblique Directions are different.
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Orthographic Projections
Most common orthographic
Projection :
Front-elevation,
Side-elevation,Plan-elevation.
Angle of projection parallel to
principal axis; projection plane
is perpendicular to axis.
Commonly used in technical
drawings
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Axonometric Orthographic Projections
Projection plane not normal to principal axis
Show several faces of the object at once
Foreshortening is uniform rather than being
related to distance Parallelism of lines is preserved
Angles are not
Distances can be measured along each principalaxis ( with scale factors )
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Isometric Projection
Most common axonometric projection Projection plane normal makes equal
angles with each axis.
i.e normal is (dx,dy,dz), |dx| = |dy|=|dz|
Only 8 directions that satisfy this
condition.
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Isometric Projection
Normal
x
z
y
Projection
Plane
y
z x
120
120
120
All 3 axes equally foreshortened
- measurements can be made
- Hence the name iso-metric
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Oblique projections.
Projection plane normal differs from the direction
of projection.
Usually the projection plane is normal to aprincipal axis.
Projection of a face parallel to this plane allows
measurement of angles and distance.
Other faces can measure distance, but not angles.
Frequently used in textbooks : easy to draw !
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Oblique projection
x
z
y
Projection
Plane
Normal
Parallel to x axis
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Geometry of Oblique Projections
Projection plane is x,y plane
L=1/tan()
- angle between normal and projectiondirection
- Determines the type of projection
is choice of horizontal angle.
Given a desired L and ,
Direction of projection is
(L.cos, L.sin,-1)
z
y
x
P
L
P=(0,0,1)
L.sin
L.cos
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Geometry of Oblique Projections
Point P=(0,0,1) maps to:
P=(l.cos, l.sin, 0) on xy plane,
and P(x,y,z) onto P(xp,yp,0)
)sin(
)cos(
lzyy
lzxx
p
p
1000
0000
0sin10
0cos01
l
l
Moband
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Mathematics of Viewing
Need to generate the transformation
matrices for perspective and parallel
projections. Should be 4x4 matrices to allow general
concatenation.
And theres still 3D clipping and moreviewing stuff to look at.
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Projections - Summary
Orthographic matrix - replace (z) axis withpoint.
Perspective matrixmultiply w by z.Clip in homogeneous coordinates.
Preserve z for hidden surface calculations.
Can find number of vanishing points.
Acknowledgments - thanks to Eric McKenzie, Edinburgh, from whoseGraphics Course some of these slides were adapted.