Hidden Line Removal
Applying vector algebra to the
problem of removing hidden lines
from wire-frame models
Convex objects
•
•
•
•
We first focus on modeling convex objects
One “outward face” cannot hide another
So visible faces can be drawn in any order
Examples: barn, cube, and dodecahedron
A non-convex object
part of this face is
hidden by that face
eye of
viewer
2D Polygons in 3D space
•
•
•
•
A polygon is a 2-dimensional figure
Its edges and corner-points are coplanar
It’s a bounded region of an infinite plane
Any plane in 3-space can be described by an
first-degree (linear) algebraic equation:
ax + by + cz = d
• Alternatively, using vector algebra, a plane can
be described using a reference-point Q and a
direction-vector N = (a, b, c), as: N • QP = 0
Face-Planes of solid objects
• Any plane surface has two sides
front
back
viewer
• When a plane is a surface of a solid object
it has an “inside” surface and an “outside”
surface (a viewer sees the “outside” one)
Angles and cosines
• An angle of 90-degrees is a “right” angle
• Angles less than 90-degrees are “acute”
• And angles over 90-degrees are “obtuse”
right angle
cosine = 0
acute angle
cosine > 0
obtuse angle
cosine < 0
Vectors and dot-products
• Vectors u = (ux, uy, uz) and v = (vx, vy, vz)
have a dot-product:
u•v = ux vx + uy vy + uz vz
• A vector’s length ║u║ equals sqrt( u•u )
• A dot-product is related to the cosine of
the angle θ between the two vectors:
u•v = ║u║║v║cosine(θ)
• So sign of dot-product tells angle’s type
Visibility of face-planes
N = outward pointing normal vector
N
D
θ
D•N > 0 (acute angle θ)
outer surface is visible
D = direction vector (from plane toward viewer’s eye)
“Hidden” face-plane
N = outward pointing normal vector
N
θ
D
D•N < 0 (obtuse angle θ)
outer surface is hidden
D = direction vector (from plane toward viewer’s eye)
Format of model’s data-set
•
•
•
•
•
Added data needed to describe our model
Number and Location of vertices as before
Edge-list is no longer needed (zero edges)
Face-list is the new information to be add
Each face is a polygon: number of sides,
list of vertices in counter-clockwise order
(as viewed from the outside of the model),
and the face-color to be used for the face
Example data-set: cube
7
0
Face-List (6 faces)
3
4
6
1
2
4-sided:
4-sided:
4-sided:
4-sided:
4-sided:
0, 1, 2, 3 (blue)
1, 0, 7, 6 (green)
2, 5, 4, 3 (cyan)
4, 5, 6, 7 (red)
6, 5, 2, 1 (magenta)
5
The vertices of each face should be listed in counterclockwise order
(an seen from the outside of the cube)
Why counterclockwise order?
• We need to compute the outward-pointing
“normal” vector for each polygonal face (to
determine if that outward face is visible)
• That normal vector is easily computed (as
a vector cross-product) if the vertices were
listed in counterclockwise order
Recall the cross-product: u×v
• If u = ( ux, uy, uz ) and v = ( vx, vy, vz ),
then w = u×v is given by these formulas:
wx = uyvz – uzvy
wy = uzvx – uxvz
wz = uxvy – uyvx
• Significance: cross-product w makes a
90-degree angle with both u and v (so
it’s normal to a plane containing u and v)
Three consecutive vertices
p×q
q
V2
V1
p = v1 – v0
q = v2 – v1
p
V0
p×q will be the outward-pointing normal vector
(if v0, v1, v2 occurred in counterclockwise order)
Demo program
• Our ‘filldemo.cpp’ application reads in the
data for a 3D model, determines which of
its polygonal faces are visible to a viewer,
fills each visible face in its specified color,
and then draws the edges of visible faces
• Datasets: plato.dat, redbarn.dat, plane.dat
• Also two models for a non-convex object:
corner1.dat and corner2.dat (same object)
In-class exercises
• Try to create the data-sets for some “more
interesting” 3D objects, (by writing a C++
program to generate the object’s vertices and
the face-lists)
• Example: An octagonal prism
–
–
–
–
Divide a circle into eight equal-size angles
Use sine and cosine to locate upper vertices
Use sine and cosine to locate lower vertices
Use number-patterns to generate its ten face-planes