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

eye of viewer

A non-convex object

part of this face is hidden by that face

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 = (u x , u y , u z ) and v = (v x , v y , v z ) have a dot-product: u •v = u x v x + u y v y + u z v z • 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

1 0

Example data-set: cube

7 3 6 4

Face-List (6 faces)

4-sided: 0, 1, 2, 3 (blue) 4-sided: 1, 0, 7, 6 (green) 4-sided: 2, 5, 4, 3 (cyan) 4-sided: 4, 5, 6, 7 (red) 4-sided: 6, 5, 2, 1 (magenta) 5 2 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 are listed in counterclockwise order

Recall the cross-product:

u ×v

• If

u

= ( u x , u y , u z ) and

v

= ( v x , v y , v z ), then

w = u ×v

is given by these formulas: w w w x y z = u y v z – u z v y = u z v x – u x v z = u x v y – u y v x • Significance: cross-product

w

makes a 90-degree angle with both

u

and

v

it’s normal to a plane containing

u

(so and

v

)

V2

Three consecutive vertices

p ×q

q V1 p

p q

= v1 – v0 = v2 – v1 V0

p ×q

will be the

outward-pointing normal vector

(if v0, v1, v2 occurred in counterclockwise order)

Remembering u

x

v

• Here’s a “trick” students use to remember the formula for computing a cross-product • It’s based on the “cofactor expansion” rule for 3x3 determinants: a b c e f d f d e d e f = a - b + c g h i h i g i g h

Applying this “trick”

i = (1,0,0), j = (0,1,0), k = (0,0,1) u = (u x ,u y ,u z ) v = (v x ,v y ,v z ) i j k u x v = u x u y u z v x v y v z = ( u y v z – u z v y , u z v x – u x v z , u x v y – u y v x )

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)

The ‘bitblit’ technique

• Our ‘filldemo’ program achieves its smooth animation, not with ‘page-flipping’, but with the so called ‘bitblit’ method • This approach has the advantage that it avoids dependency on Radeon hardware (e.g., using the CRTC_START register)

How ‘bitblit’ is used

CRTC_START

(never gets changed)

All the drawing is done to page 1 (which never gets displayed) Then after an image has is fully drawn, it is rapidly copied to page 0 (‘bitblit’) Visible page (page 0) Drawing page (page 1) VRAM

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