CSC 308 Graphics Programming

1
CSC 308 Graphics Programming

• 3D Graphics
• Information modified from Fergusons Computer
  Graphics via Java, and
• Fundamentals of Computer Graphics. by Shirley

Dr. Paige H. Meeker Computer Science Presbyterian
College, Clinton, SC
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Leaving the world of 2D

• What have we learned?
• Data structures to store various types of
  graphical objects (points, lines, shapes,
  drawings)
• How to draw these on the screen for the viewer to
  see
• How to make our objects move, rotate, and
  grow/shrink even reflect and shear.
• Touched on making curves
• Accept user input via keyboard or mouse
• Animate our shapes.

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Thus,

• We have mastered most of the techniques required
  for 2d graphics.
• Could now write applications as diverse as
  mapping systems and video games.
• We live in the world of 3d not 2d what can we
  take from what we have already learned about 2d
  and apply it to our next frontier?
• There is something almost magical about staring
  through a computer screen into a cyberspace and
  seeing worlds and objects that dont really
  exist. Lets get started!

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Introduction

• Modelling 3d with data structures
• Expanding 2d points,lines, transforms etc.
• Drawing 3d on a 2d screen
• projections

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3D coordinate system

• In 2d we had height and breadth - but all shapes
  were flat
• It makes sense to add a third coordinate to
  represent depth
• By convention we use 3 mutually perpendicular
  axes, ( any 3 non co-planar axes would do).
• Any point in 3d space can thus be represented by
  just those three numbers x,y z

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2D -gt 3D

• representation of a cube using our new coordinate
  system.
• Each line of the cube (a-l) can still be
  described by its endpoints just as in 2d, but
  each point now has 3 coordinates.
• good news is that we can still use our drawing,
  shape, line and point classes.
• bad news is that they are all going to require
  some (hopefully small) alterations to work in 3d.

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3D coordinate system
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3D Points

• Still using homogenous coordinates
• Now need 4
• Why?

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3D Transformations

• Since we now have an additional coordinate to
  cope with, our transformation matrices must also
  expand to cope.
• Each transformation matrix with require an extra
  row and an extra column to become 4x4 matrices.
• The change in Scaling and Translating is very
  straightforward. Rotation is a little more
  tricky (as usual!)

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Scaling
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Translation
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Rotation (about axes)

• In 2 dimensions, rotation were defined to take
  place about some stationary point - the centre of
  rotation.
• In 3d, all rotation take place about one of the
  axes.

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Rotation (about axes)
-90 about x axis
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Rotation

• x-axis

• y-axis

• z-axis

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Local Rotation

• translate the object to be rotated to the origin,
  perform the rotation and translate it back to
  where it started from

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Modify the Point Class

• Using Point2d as a guide, how would you now
  modify it to make a class Point3d.java?

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How about Line, Drawing, and Shape?

• How would these change from 2d to 3d?

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Viewing

• The really ironic thing about 3d graphics is that
  currently, we only have a 2d screen to display
  them on. So, after all the work involved in
  calculating the location of the points, lines,
  shapes, etc, we then have to figure out how to
  draw them on a 2d screen so that they still
  retain their 3d appearance!

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2D Cartesian Coordinate System
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3D Cartesian Coordinate System
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Handed Coordinate Systems

• Which way is up?

Right Handed
Left Handed
NOTE Rotating the system does not change it's
directionality in relation to itself. Various
industries will take advantage of this to orient
a the system in a convenient way.
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Right Handed Rotation Example
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Polar (2d) and Spherical(3d)Coordinate Systems
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Projection

• In general, projections transform points in a
  coordinate system of dimension n into points in a
  coordinate system of dimension less than n. In
  fact, computer graphics has long been used for
  studying n-dimensional objects by projecting them
  into 2D for viewing.

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Projection

• The projection of a 3D object is defined by
  straight projection rays (called projectors)
  emanating from a center of projection, passing
  through each point of the object, and
  intersecting a projection plane to form the
  projection.

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Projection

• Representation of a 3d object on a 2d screen
• Several different projections exist
• Well take a look at some common projections

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Terms and definitions
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Terms and Definitions

• To work properly, the user has to position the
  center of projection (aka eye) in order to
  achieve good view quality
• Here, there is only 1 eye humans have two. By
  producing 2 simultaneous projections based on two
  slightly different centers of projection (about 8
  cm apart), and feeding them separately to the
  left and right eyes, an imitation of 3d can be
  obtained (think VR goggles).

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so?

• We need to calculate the screen coordinates
  (xsys) of each projected point from the 3d
  coordinates
• (theyre not the same!)
• Usually!
• Fortunately we can consider this calculation as
  a transformation.
• Can use matrices to represent the various ways of
  doing it.

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Projections

• Two basic classes of projections
• Perspective distance from center of projection
  to the projection plane is finite. Causes the
  size of the perspective projection of an object
  to vary inversely with the distance of that
  object from the center of projection.
• Parallel distance from center of projection to
  the projection plane is infinite. (Better for
  exact measurements less realistic.)

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Orthographic and PerspectiveViewing Projections
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Parallel Projection
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Perspective Projection
d
z
C
(xs,ys)
ps
By similar triangles
p
(x,y,z)

screen
x
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Perspective Projection
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Perspective Projection
So. Divide by
want
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Now

• In addition to the perspective transformation, we
  may also need to apply a viewpoint transformation
  before displaying data on the screen. This give
  the camera (eye point) the ability to move, pan,
  tilt and zoom.

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Default viewpoint
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Alternative viewpoint
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Viewpoint translation and rotation

• Moving the viewpoint by viewpoint transformation
  allows us to describe the position of the camera
  in terms a translation of its position and a
  rotation of its orientation.
• A (camera or viewpoint) is described as having
  six degrees of freedom i.e. we can translate it
  in the x, y and z directions and we can spin it
  about the x,y, and z axes.

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Viewpoint translation
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Viewpoint rotation
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How do we do it?

• Already have all the mathematical tools required
  - matrices from the previous section.
• Dont actually move the camera,
• Transform the world (or the model of it) in
  precisely the opposite sense to the way we would
  have moved the camera.
• These operations are in fact equivalent - you end
  up with the same view.
• Easier to program that way.
• Think of it in terms of an idle cameraman who has
  the world moved instead of shifting the camera

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Moving the camera or moving the world

• Viewpoint transformation is itself a combination
  of 2 separate transformations,
• one for the rotation,
• one for the translation.
• These two transformations must be concatenated to
  produce the complete transformation. It is
  important to note that the rotation represents a
  local rotation of the entire world about the
  position of the camera and must be applied first.

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Implementing 3d

• Consider how implement data model
• Consider overall system design
• Consider changes to 2d classes
• Consider how to implement projection/viewpoint
  transformation
• Consider changes to 2d classes

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Implementing 3d

• Changes to the 2d classes
• Matrix class no change at all.
• Point3d
• Add an extra (z) coordinate
• change constructor, toString and setters and
  getters to match.

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Implementing 3d

• Line3d
• A constructor needs altering to allow the
  specification of the z coordinate.
• May need to alter the drawing method as well

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Implementing3d

• Tranformation3d
• Increase size of matrices to 4x4
• Add all of the matrices for rotation round the
  axes (rotatex, rotatey, rotatez)
• Add matrices for various projections

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To display, what we need to do is

• Take the 3d data model that is stored in our
  shapes, lines and points - and apply a series of
  transformations to it before somehow plotting it
  on the screen.

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Those transformations are

• a local rotation of the model about the
  viewpoints x axis
• a local rotation of the model about the
  viewpoints y axis
• a local rotation of the model about the
  viewpoints z axis
• a translation of the model in the x direction by
  an amount equal to the (negative) displacement of
  the viewpoint from the origin in the x direction
• a translation of the model in the y direction by
  an amount equal to the (negative) displacement of
  the viewpoint from the origin in the y direction
• a translation of the model in the z direction by
  an amount equal to the (negative) displacement of
  the viewpoint from the origin in the z direction
• one of the projection transformations

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• Concatenate them (multiply them in the order
  given)
• Dont transform the model!
• Transform a copy of each point as you come to use
  it
• I.e. in the draw() methods

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Concatenate the transformations

• The rotation and translation transforms can be
  concatenated into one viewpoint transformation in
  the Gapp class.

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Sharing the transformations

• Once the concatenation has been done, Gapp calls
  the setDrawingTransformation method of the
  drawing objects and thus sets the transformation
  to be used by the entire data structure.

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Using the drawing transformation

• It is still the draw methods of the line class
  that does the actual work of line drawing. (and
  later in the Bezier3d and Spline3d we will be
  adding)
• The projection transformation is dealt with
  entirely within the Line3d class (and later in
  the Bezier3d and Spline3d we will be adding)
• Make copies of the endpoints of the line, apply
  the drawing transformation to those copies
  (LEAVING THE ORIGINALS UNCHANGED) and use the x
  and y coordinates of those points as the
  endpoints of the lines we draw on screen.

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Summary

• Extra (z) coordinate
• Rewrite transformations as 4x4 matrices
• Projection reducing 3d model to 2d display
• Viewpoint transformation
• Six degrees of freedom of camera
• Implementation of 3d graphics

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Homework FIRST LARGE PROJECT!!

• Alter the Point, Line, Shape, and Transformation
  classes to be 3d instead of 2d.
• Add a perspective and parallel projection method
  to the Transformation3d class.
• Enjoy Fall Break.
• Next Wednesday Well explain in more detail how
  to implement the viewing transform and display
  our graphics and define our user interface.
• DUE Monday, October 30 A working 3D Modeler
  that will allow the user to create 3d shapes,
  apply transformation to those shapes, and view
  them in a parallel or perspective trasformation.