Hidden Line Removal

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Hidden Line Removal

• Graphics Lecture 11

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Introduction

• Depth cueing
• Surfaces
• Vectors/normals
• Hidden face culling
• Convex/concave solids

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Perspective Confusion
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Hidden Line Removal
(show demo basic3d wireframe surface)?
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Hidden Line Removal

• No one best algorithm
• Look at a simple approach for convex solids
• based upon working out which way a surface is
  pointing relative to the viewer.

convex
concave
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Based on surfaces not lines

• Need a Surface data structure
• WireframeSurface
• made up of lines

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Flat surfaces

• Key requirement of our surfaces is that they are
  FLAT.
• Easiest way to ensure this is by using only three
  points to define the surface.
• (any triangle MUST be flat - think about it)
• but as long as you promise not to do anything
  that will bend a flat surface, we can allow them
  to be defined by as many points as you like.
• Single sided

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Which way does a surface point?

• Vector mathematics defines the concept of a
  surfaces normal vector.
• A surfaces normal vector is simply an arrow that
  is perpendicular to that surface (i.e. it sticks
  straight out)?

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Determining visibility

• Consider the six faces of a cube and their normal
  vectors
• Vectors N1 and N2 are are the normals to surfaces
  1 and 2 respectively.
• Vector L points from surface 1 to the viewpoint.
• It can be seen that surface 1 is visible to the
  viewer whilst surface 2 cannot be seen from that
  position.

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Determining visibility

• Mathematically, a surface is visible from the
  position given by L if
• Where ? is the angle between L and N.
• Equivalently,

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Determining visibility

• Fortunately we can calculate cos ??from the
  direction of L (lx,ly,lz) and N (nx,ny,nz)
• This is due to the well known result in vector
  mathematics - the dot product (or the scalar
  product) whereby

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Determining visibility

• Alternatively
• Where L and N are unit vectors (i.e of length 1)?

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How do we work out L.N?
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How do we work out L.N?

• At this point we know
• we need to calculate cos ?
• Values for lx,ly,lz
• The only things we are missing are nx,ny,nz

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Calculating the normal vector

• If you multiply any two vectors using the vector
  product, the result is another vector that is
  perpendicular to the plane (i.e normal) which
  contained the two original vectors.

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IMPORTANT

• We need to adopt the convention that the
  calculated normal vector points away from the
  observer when the angle between the two initial
  vectors is measured in an clockwise direction.
• Failure to do this will lead to MAJOR confusion
  when you try and implement this
• Believe me, I know.

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Calculating the normal

• Where to find two vectors that we can multiply?
• Answer we can manufacture them artificially from
  the points that define the plane we want the
  normal of

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Calculating the normal

• By subtracting the coordinates of consecutive
  points we can form vectors which a guaranteed to
  lie in the plane of the surface under
  consideration.

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Calculating the normal

• We define the vectors to be anti-clockwise, when
  viewing the surface from the interior
• (imagine the surface is part of a cube and your
  looking at it from INSIDE the cube).
• Following the anticlockwise convention mentioned
  above we have produced what is known as an
  outward normal.

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IMPORTANT

• An important consequence of this is that when you
  define the points that define a surface in a
  program, you MUST add them in anti-clockwise order

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Calculating the normal
This is a definition of the vector product
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Visibility

• At this point we know
• we need to calculate cos ?
• Values for lx,ly,lz
• values for nx,ny,nz

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Visibility

• If
• then draw the surface
• else
• dont! (or draw dashes)?

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Making it easier

• Actually, in certain cases we can simplify things
  a little. If the viewpoint lies somewhere on the
  negative side of z-axis, as it did when we first
  set up the projection transformations (i.e
  without any viewpoint transformations) we can
  forget about L and cos ?
• All we really need to know is whether or not the
  normal points into the screen or out of it, i.e.
  is nz positive or negative? In that case, all we
  need to do is calculate nz and test it

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An alternative approach

• Consider where the extension of the surface cuts
  the z-axis

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More complex shapes
Concave objects
Multiple objects
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More complex shapes

• In these cases, each surface must be considered
  individually. Two different types of approach are
  possible
• Object space algorithms - examine each face in
  space to determine it visibility
• Image space algorithms - at each screen pixel
  position, determine which face element is
  visible.
• Approximately, the relative efficiency of an
  image space algorithm increases with the
  complexity of the scene being represented, but
  often the drawing can be simplified for convex
  objects by removing surfaces which are invisible
  even for a single object.

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The z-buffer (or painters) algorithm

• based upon sorting the surfaces by their
  z-coordinates. The algorithm can be summarised
  thus
• Sort the surfaces into order of increasing depth.
  Define the maximum z value of the surface and the
  z-extent.
• resolve any depth ambiguities
• draw all the surfaces starting with the largest
  z-value

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Ambiguities

• Ambiguities arise when the z-extents of two
  surfaces overlap.

lttwo books demogt
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Ambiguities front view
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Resolving Ambiguities

• An algorithm exists for ambiguity resolution
• Where two shapes P and Q have overlapping
  z-extents, perform the following 5 tests (in
  sequence of increasing complexity).
• If any test fails, draw P first.

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x - extents overlap?
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y -extents overlap?
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Is Q not completely on the side of P nearest the
viewer?
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Is P not completely on the side of Q further from
the viewer?
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Does the projection of the two surfaces overlap?
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If all tests are passed

• then reverse P and Q in the list of surfaces
  sorted by Zmax
• set a flag to say that the test has been perfomed
  once.
• The flag is necessary for the case of
  intersecting planes.
• If the tests are all passed a second time, then
  it is necessary to split the surfaces and repeat
  the algorithm on the 4 surfaces

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End up drawing Q2,P1,P2,Q1
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Summary

• Need for depth cues and hidden line removal
• Using surfaces rather than lines
• Calculating which way a surface is facing
  (surface normal)?
• Base a decision on visibility on normal vector
  and observer vector

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Next time.

• Rendering
• Shading/Colour
• Lambert/Gouraud/Phong