Basic graphics

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Basic graphics
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Review

• Viewing Process, Window and viewport, World,
  normalized and device coordinates
• Input and output primitives and their attributes.
  Rendering. Vector and raster displays
• Basic graphics in an universal programming system
• Vector Algebra
• Homogeneous Coordinates and Transformation of
  Space

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Preliminaries

• What newcomers to Shape Modeling and Computer
  Graphics have to know
• Fundamental ideas and vocabulary in
• Geometrical Modeling and Computer graphics
• Graphics and geometry APIs
• Transformations
• Curves and surfaces design
• Rendering technique
• Physics-based approaches for modeling

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Overview

• Are they shadows that we see?
• And can shadows pleasure give?
• Pleasures only shadows be
• Cast by bodies we conceive
• And are made the things we deem
• In those figures which they seem
• From Are They Shadows by Samuel Daniel

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3D Viewing

• Many of the 3D images seen on a screen are really
  a trick of the eye. The modeling is performed in
  3D, but the rendering has to be converted to 2D

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3D Viewing
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3D Viewing
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3D Viewing

• A useful metaphor for creating 3D scenes is the
  motion of a synthetic camera
• We imagine that we can move our camera to any
  location, orient it in any way we wish, and, with
  a snap of the shutter, create a 2D image
  (photo) of a 3D object
• The camera, of course, is really just a computer
  program that produces an image on a display
  screen

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3D Viewing

• Rendering is the process by which a computer
  creates images from models
• The rendering of 3D pictures is accomplished
  through projection draw a line from the eye (P0)
  with a straight line to the (x,y,z)

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3D Viewing

• Creation of our photo is accomplished as a
  series of steps
• Specification of projection type to transform 3D
  objects onto a 2D projection plane
• Specification of viewing parameters. We must
  specify the conditions under which we want to
  view the 3D real world dataset
• Clipping in three dimensions. We must ignore
  parts of the scene that are not candidates for
  ultimate display

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3D Viewing

• Given that output primitives are specified in
  world coordinates, the graphics subroutine
  package must be told how to map world coordinates
  onto screen coordinates or device coordinates
• So if we have world-coordinate window, and a
  corresponding rectangular region in screen
  coordinates, called the viewport, we have to
  produce the transformation that maps the window
  into the viewport

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3D Viewing

• We may want to ignore parts of the scene that are
  behind us or too far distant to be clearly
  visible
• The view volume bounds that portion of the world
  that is to be clipped out and projected onto the
  view plane

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3D Viewing

• In the output pipeline, the application program
  takes description of objects in terms of
  primitives and attributes stored in or derived
  from an application model or data structure, and
  specifies them to the graphics package, which in
  turn clips and converts them to the pixels seen
  on the screen
• CG packages supports a basic collection of
  primitives lines, polygons, circles and
  ellipses, and text

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3D Viewing

• Raster displays store the display primitives in a
  refresh buffer in terms of the primitives
  component pixel
• Pixels are points of a rectangular array or the
  smallest visible elements that the display
  hardware can put on the screen

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Basic graphics in an universal programming
system (OpenGL)

• The OpenGL graphics system is a software
  interface to graphics hardware
• It allows you to create interactive programs that
  produce color images of moving three-dimensional
  objects
• With OpenGL, you must build up your desired
  model from a small set of geometric primitives
  points, lines, and polygons

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Basic graphics in an universal programming
system (OpenGL)

• The OpenGL Utility Library (GLU) provides many of
  the modeling features, such as quadric surfaces
  and NURBS
• OpenGL is a state machine. You put it into
  various states that then remain in effect until
  you change them
• Many state variables refer to modes that are
  enabled or disabled

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Basic graphics in an universal programming
system (OpenGL)

• All data, whether it describes geometry or
  pixels, can be saved in a display list for
  current or later use
• All geometric primitives are eventually described
  by vertices. Parametric curves and surfaces may
  be initially described by control points.
  Evaluators provide a method to derive the
  vertices
• You can handle input events and register callback
  command that are invoked when specified events
  occur, for instance when the mouse is moved while
  a mouse button is also pressed

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Basic graphics in an universal programming
system (OpenGL)

• One of the most exciting things you can do on a
  graphical computer is draw pictures that move, or
  animate objects
• Most OpenGL implementation provide
  double-buffering hardware or software that
  supplies two complete color buffers. Once is
  displayed while the other is being drawn. When
  the drawing of a frame is complete, the two
  buffers are swapped

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Vector Algebra

• A vector is a mathematical entity with two
  attributes direction and magnitude. The
  magnitude of vector P (x,y,z) is P
• The direction of a vector can be expressed by the
  cosines of the angles it makes with the
  coordinate axis x/P, y/ P, z/P.
• The three unit vectors in the directions of the
  coordinate axes are traditionally denoted
• i (1,0,0), j (0,1,0) and k (0,0,1).

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Vector Algebra

• Vectors permit two fundamental operations You
  can add them and you can multiply them by
  scalars.
• Let
• a1 a1i, a2 a2j, a3 a3k, and
• b1 b1i, b2 b2j, b3 b3k. We can write
• a b (a1 b1)i (a2 b2)j (a3 b3)k ,
• ?a ?a1 i ?a2 j ?a3k .
• To form a linear combination of two vectors, a
  and b, we scale each of them by some scalars,
  say, ? and ? and add the weighted versions to
  form new vector, ?a ?b.

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Vector Algebra

• The dot product of two vectors. To calculate the
  dot product we multiply corresponding components
  of the vectors and add the result.
• The cross product (also called the vector
  product) of two vectors is another vector. Given
  the 3D vectors
• a (ax,ay,az), and b (bx,by,bz),
• their cross product is denoted as a?b or a?b.
  It is defined in terms of the standard unit
  vector i,j, and k by
• a?b (axbz azby)i (azbx axbz)j (axby
  aybx)k.

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Vector Algebra

• Vector equation of line. Suppose we are given a
  point P0 and a unit vector u along the line L.
  Give the names r0 and r to the vectors OP0 and
  OP connecting the point O with the point P0 and
  an arbitrary point P, respectively. Then the
  vector P0P r r0 coincides with the direction
  of the vector u, so
• r r0 ? u. And we have
• r r0 ? u.

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Vector Algebra

• Vector differentiation. If a vector a is the
  function of time t, a(t) a1(t) i a2(t) j
  a3(t) k, then
• i j k.

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Homogeneous Coordinates and Transformation of
Space

• Each primitive has a mathematical representation
  which can be expressed as a data or type
  structure fore storage and manipulation by a
  computer
• Applying transformations to graphical primitives
  can be done by a process called instancing

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Projections

• 3D objects can be projected on 2D view plane
• There are two basic methods
• parallel projection
• perspective projection
• The parallel projection can be considered as a
  special case of of perspective projection

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Projections

• Orthographic Parallel Projections
• The most common type of orthographic projection
  is the front-elevation

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Homogeneous Coordinates and Transformation of
Space

• A point (x, y, z) in three dimensional Cartesian
  space is represented in R4 by the vector (x, y,
  z, 1), or by any multiple (rx, ry, rz,r) (with, r
  ? 0). When W ? 0, the homogeneous coordinates (X,
  Y. Z. W) represent the Cartesian point (x, y, z
  (X/W, Y/W, Z/W)
• A point of the form (X, Y, Z,0) does not
  correspond to a Cartesian point, but represents
  the point at infinity in the direction of the
  three-dimensional vector (X, Y, Z). The set of
  all homogeneous coordinates (X, Y, Z. W) is
  called (three-dimensional) projective space and
  denoted P3

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Homogeneous Coordinates and Transformation of
Space

• A transformation of projective space is a
  mapping L P3 ? P3 of the form
• The 4?4 matrix M is called the homogeneous
  transformation matrix of L. If M is a
  non-singular matrix then L is called a
  non-singular transformation. If m14 m24 m34
  0 and m44 ? 0, then L is said to be an affine
  transformation. (Affine transformations
  correspond to translations, scalings, rotations
  etc. of three-dimensional Cartesian space.)
• A point of the form (X, Y, Z,0) does not
  correspond to a Cartesian point, but represents
  the point at infiniy

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Homogeneous Coordinates and Transformation of
Space

• Translations
• - The transformation matrix of a translation by
  x0, y0, and z0 units in the x-, y- and z-
  directions respectively, is
• - Hence, P(x, y, z) is transformed to P'(x x0,
  y y0, z z0)

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Homogeneous Coordinates and Transformation of
Space

• Scalings and Reflections
• - A scaling about the origin by a factor sx, sy,
  and sz, in the x-, y-, and z-direction ,is
  respectively, obtained by the following
  transformation matrix
• - The transformation matrices of the reflections
  Ryz in the x 0 plane, Rxz, in the y 0 plane,
  and Rxy in the z 0 plane, are obtained by
  taking a scaling of -1 in one of the coordinate
  directions

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Homogeneous Coordinates and Transformation of
Space

• Rotations about the Coordinate Axes
• Rotations in space take place about a line
  called its rotation axis
• - The rotations about the three coordinate axes
  are called the primary rotations
• 1. Rotation about the x-axis through an angle ?x
• 2. Rotation about the y-axis through an angle ?y
• 3. Rotation about the z-axis through an angle ?z

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Homogeneous Coordinates and Transformation of
Space

• Rotation about an Arbitrary Line
• -Rotation through an angle ? about an arbitrary
  rotation axis is obtained by transforming the
  rotation axis to one of the coordinate axes,
  applying a primary rotation through an angle ?
  about the coordinate axis, and applying the
  transformation which maps the coordinate axis
  back to the rotation axis

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Homogeneous Coordinates and Transformation of
Space

• Rotation about an Arbitrary Line
• Apply the translation T(-pl,-p2 -p3) which maps P
  to the origin and the rotation axis to the line
  OR where R(r1,r2,r3) U
• Suppose r2 and r3 are not both zero. Apply a
  rotation through all angle ?x, about the x-axis
  so that the line OR is mapped into the xz-plane
• Apply a rotation about the y-axis so that the
  line OR' is mapped to the z-axis.
• Apply a rotation through an angle ? about the
  z-axis (Figure 2(c)).
• Apply the inverses of the transformations 1-3 in
  reverse order