COMPUTER GRAPHICS

USING OPEN GL SECOND EDITION

F. S. HILL, JR.

2.3.5 Filling Polygons

CHAPTER 4: Vector Tools for Graphics

4.2

  REVIEW OF VECTORS

• Geometrically, vectors are objects having length and direction. They represent various physical entities such as force, displacement, and velocity.

4.2.1 Operation with Vectors

• Vectors can be added and they can be multiplied by a scalar. If a = (2, 5, 6), b = (-2, 7, 1) and scalar s = 6: a + b = (0, 12, 7) and sa = (12, 30, 36).

4.2.2 Linear Combinations of Vectors

• A linear combination of m vectors v₁, v₂, ..., v_m, is a vector of the form
  w = a₁v₁, a₂v₂, ..., a_mv_m       where a₁, a₂, ..., a_m are scalars.
• An affine combination is a restriction on a linear combination of vectors in that the scalers or coefficients a₁, a₂, ..., a_m must add up to unity: a₁ + a₂ + ... + a_m = 1.
• A convex combination is a further restriction in that the coefficients must be nonnegative. Therefore, 0 ≤ a_i ≤ 1 for i = 1, ..., m.
  • 0.3a + 0.7b is convex.
  • The set of all convex combinations is v = (1 - a)v₁ + av₂        for all v₁ and v₂.
  • For three vectors,       q = a₁v₁ + a₂v₂ + (1 - a₁ - a₂)v₃        where a₁ plus a₂ does not exceed unity.
  • A convex linear combination of two vectors defines a line, three vectors determines a triangular plane.

4.2.3 The Magnitude of a Vector; Unit Vectors

• By the Pythagorean theorem     ______________________
  the magnitude of vector |w| = √ w₁² + w₂² + ... + w_n²

  Normalizing a vector means scaling it to a length of unity,

  â =

  a  |a|

• Example: if a = ( 3, -4 ), then | a | = ( 3², -4² )^½ = 5 and â = ( ³ / ₅, ^-4 / ₅ ). This unit vector is a direction.

4.3

  THE DOT PRODUCT

• The dot product of two vectors, (a₁, a₂) and (b₁, b₂), is the scalar a₁b₁ + a₂b₂. For example, the dot product of:
  • (3, 4) and (1, 6) = 3 × 1 + 4 × 6 = 3 + 24 = 27
  • (2, 3) and (9, -6) = 2 × 9 + 3 × -6 = 18 - 18 = 0
• The general definition of the dot product:
  • The dot product d of two n-dimensional vectors
    v = (v₁, v₂, ... v_n) and w = (w₁, w₂, ... w_n) is denoted as v • w and has the value
    

    d = v • w =	n	viwi
    	∑
    	i=1

  • Example, the dot product of
    • (2, 3, 1) and (0, 4, -1) = 2 × 0 + 3 × 4 + 1 × -1 = 0 + 12 - 1 = 11.
    • (2, 2, 2, 2) • (4, 1, 2, 1.1) = 2 × 4 + 2 × 1 + 2 × 2 + 2 × 1.1 = 8 + 2 + 4 + 2.2 = 16.2.

4.3.2 The Angle Between Two Vectors

• For vectors b and c at angles Φ_b and Φ_c:
      b = X_b, Y_b
        = |b| cos Φ_b, |b| sin Φ_b
  and
      c = |c| cos Φ_c, |c| sin Φ_c
• The dot product of these is
    b•c = |b||c| cos Φ_c cos Φ_b + |b||c| sin Φ_b sin Φ_c
        = |b||c| cos (Φ_c - Φ_b)
• We now have the angle between them
    b•c = |b||c| cos (θ)
• Divide both sides by |b||c| and use the unit vector notation
          _^
      b = b/|b| to obtain
               _^ ^
      cos(&theta) = b•c
• EXAMPLE: Find the angle between b = (3, 4) and c = (5, 2).
                              
    |b| = √3²+4² = √9+16 = √25 = 5
                              
    |c| = √5²+2² = √25+4 = √29 = 5.385
      _^
      b = (3/5, 4/5) = (0.60, 0.80)
      _^
      c = (0.9285, 0.3714)
    _^ ^
    b•c = 0.6 × 0.9285 + 0.8 × 0.3714 = 0.5571 + 0.2971 = 0.8542 = cos(θ)
      θ = 31.3287°.

4.3.3 The Sign of b • c, and Perpendicularity

• Cos(θ) is positive if |θ| less than 90°, zero if |θ| equal to 90°, and negative if |θ| greater than 90°.
• DEFINITION: Vectors b and c are perpendicular if b•c = 0.
• Other names for perpendicular are orthogonal and normal.
• The 2D vectors (1, 0) and (0, 1) are mutually perpendicular unit vectors.
• DEFINITION: The standard unit vectors in 3D have components
    i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1).
• Using these definitions we can write any 3D vector such as (a, b, c) = ai + bj + ck.
• EXAMPLE:     v = (2, 5, -1)
  is the same as   2(1, 0, 0) + 5(0, 1, 0) - 1(0, 0, 1)
                 = 2i + 5j - k

4.3.4 The 2D "Perp" Vector

• Suppose a vector a has components (a_x, a_y). If you exchange the x and y components, negate one of them and dot this to the original, the result is zero.
     (a_x, a_y) • (-a_y, a_x) = 0
  
• This is also a•a^⊥ = 0
• For convenience we use the symbol ⊥ (pronounced "perp") to designate a normal or perpendicular angle.
• DEFINITION: Let a = (a_x,a_y). Then a^⊥ = (-a_y,a_x) is the counterclockwise perpendicular to a.

4.3.5 Orthoganal Projections and the Distance from a Point to a Line

• Given two points A and C and a vector v, how do you decompose it into components of length K and M? One solution is
      c = Kv + Mv^⊥.
• To solve for two unknowns you have to set one variable to zero. If we multiply both sides by v we get
    c•v = Kv•v + Mv^⊥•v. Now Mv^⊥•v = 0 so we get

  K =	c•v
  	v•v

• Similarly if we "dot" both sides with v^⊥ we get

  M =	c • v⊥
  	v • v

• Putting it all together we have

  c = (

  v • c |v|²

  )v + (

  v⊥•c |v|²

  )v⊥

• The part along v is know as the orthogonal projection of c onto the vector v.
• Further, the second term, the distance from C to vector v

  distance = |

  v⊥•c |v|²

  v⊥| =

  | v⊥•c | |v|

  =

  |v⊥•(C-A)| |v|

4.4

  THE CROSS PRODUCT OF TWO VECTORS

• The cross product or the vector product of two vectors is another vector that is perpendicular to both the given vectors. It is only defined for three-dimensional vectors.
• Given the 3D vectors a = (a_x, a_y, a_z ) and b = (b_x, b_y, b_z ), their cross product is denoted a × b.
• It is defined in terms of the standard unit vectors, i, j, and k:
  a × b = ( a_yb_z - a_zb_y )i + ( a_zb_x - a_xb_z )j + ( a_xb_y - a_yb_x )k
• As this is difficult to remember it is often written as

  	|	i	j	k	|
  a × b =		ax	ay	az
  		bx	by	bz

4.5.1 Coordinate Systems and Coordinate Frames

• A coordinate frame consists of a specific point, γ, called the origin, and three mutually perpendicular unit vectors (axes), a, b, and c. This frame resides at some point γ within the "world".
• To represent the vector v, we find three numbers ( v₁, v₂, v₃ ) such that
  v = v₁a, v₂b, v₃c
  and say that v "has the representation" (v₁, v₂, v₃ ) in this system.
• To represent a point P, we view its location as an offset from the origin by a certain amount.
  P = γ + p₁a, p₂b, p₃c
• The homogeneous representation of graphics appends a 0 as the fourth component of a vector and a 1 as the fourth component of a point.
• Using matrix multiplication we can formally write any v and P as

  v = ( a, b, c, γ )	(	v1	)
  		v2
  		v3
  		0

  P = ( a, b, c, γ )	(	P1	)
  		P2
  		P3
  		1

4.5.2 Affine Combinations of Points

• Consider forming a linear combination of two points P and R using scalars ƒ and g. We have
  E = ƒP + gR
  If ƒ and g are not affine (sum to unity) and we move between coordinate frames, the sum, E', does not have uniform values and distortion results.
  If E = ƒP + gR is shifted by u, P is shifted to P + u, R is shifted to R + u and E' = E + u. Instead we have
  E = ƒP + gR + (ƒ + g)u
• Another interesting thing is to form a point A offset by vector v and scale it to t. If v is the difference between another point, v = B - A
  P = A + t( B - A )
    = tB + ( 1 - t )A

• The centroid (center of gravity) of triangle D, E, F is	C =	D + E + F
  		3

4.5.3 Linear Interpolation of Two Points

GLdouble lerp( GLdouble a, GLdouble b, GLdouble t )
{   return a + ( b - a ) * t;
}

Point3D tween( Point3D A, Point3D B, GLdouble t )
{	Point3D tmp = new Point3D;
	tmp.x = lerp( A.x, B.x, t );
	tmp.y = lerp( A.y, B.y, t );
	tmp.z = lerp( A.z, B.z, t );
	return tmp;
}

4.5.4 "Tweening" for Art and Animation

void drawTween( Point3D A[ ], Point3D B[ ], int n, float t )
{	for( int i = 0; i < n; i++ )
	{	Point3D P;
		P = tween( A[ i ], B[ i ], t );
		if( i == 0 )
			moveTo( P.x, P.y );
		else
			lineTo( P.x, P.y );
	}
}

4.5.6 Representing Lines and Planes

• A line is defined by two points, say, C and B. It is infinite in length, passing through the points and extending forever in both directions. A line segment, or segment for short, is defined by its endpoints. It extends only from one to the other. Its parent line is the infinite line that passes through its endpoints. A ray is "semi-infinite". It is specified by a point and a direction, starting at that point and extending infinitely far in a given direction.
• In computer graphics the most important representaion of lines is the parametric form. As the parameter t varies, the point P trace out all points on the straight line defined by C and B. The line is L and b = B - C.
  L(t) = C + bt.
• To develop the point normal or implicit form of a line we start with a familiar form,
  fx + gy = 1
  where f and g are constants. Every point ( x, y ) that satisfies this equation is on the line. Unfortunatly, a 3D line requires two equations.
  • This can also be written as a dot product,
    ( f, g ) • ( x, y ) = 1.
    For every point on a line a certain dot product must have the same value.
  • Suppose we know that line L passes through points C and B. If we can find a vector n that is perpendicular to L, then for any point R = ( x, y ) on L, the vetor R - C must be perpendicular to n. We have the following condition on R:
    n • ( R - C ) = 0.
    This is the point normal equation for the line, expressing the fact that a certain dot product must turn out to be zero for every point R on the line.
  • To find a suitable n, let b = R - C denote the vector from C to B. b^⊥ will serve as the desired n, as will any scalar of b^⊥.
    If C = ( 3, 4 ) and B = ( 5, -2 ), then b = B - C = ( 2, -6 ) and b^⊥ = ( 6, 2 ). Choosing C as the point on the line,
    ( 6, 2 ) • ( ( x, y ) - ( 3, 4 ) ) = 0
    ( 6x + 2y ) - ( 6×3 + 2×4 ) = 0
    6x + 2y = 26
• Object faces are polygons that lie in parent planes. Planes also have three forms.
  • Three points in the plane, C, B, and A.
  • The parametric form consists of a point C and two nonparallel vectors a and b. If we are given three noncollinear points A, B, and C in the plane, we take a = A - C and b = B - C. The parametric form for this plane is the point C plus any two multiples of a and b.
    P( s, t ) = C + as + bt
  • A plane is specified by any point B = ( b_x, b_y, b_z ), and the direction n = ( n_x, n_y, n_z ) normal or perpendicular to the plane. For any point R = ( x, y, z ) in the plane, the vector from R to B must be perpendicular to n:
    n • ( R - B ) = 0.
    This is the point normal equation of the plane.
    If plane P passes through ( 1, 2, 3 ) with normal vector ( 2, -1, -2 ), the point normal is
    ( 2, -1, -2 ) • ( ( x, y, z ) - ( 1, 2, 3 ) ) = 0
    ( 2x - y - 2z ) - ( 2×1 - 1×2 - 2×3 ) = 0
    2x - y - 2z = -6

4.6

  FINDING THE INTERSECTION OF TWO LINE SEGMENTS

One line segment has endpoints A and B, the other C and D. Let b = B - A. Then
AB( t ) = A + bt.
and if d = D - C,
CD( u ) = C + du.
The parametere t and u must be different. For the parent lines to intersect there must be specific values of t and u for which the two equations are equal:
A + bt = C + du.
Define c = C - A.
bt = c + du.
Now we have two equations in two unknowns. If we dot both sides with d^⊥ to eliminate the term in d.
d^⊥ • bt = d^⊥ • c + d^⊥ • du
d^⊥ • bt = d^⊥ • c + 0
d^⊥ • bt = d^⊥ • c
If d^⊥ • b is zero, the lines are parallel. If it is not zero

t =	d⊥ • c
	d⊥ • b

Then we dot both sides with b^⊥.

u =	b⊥ • c		or, as b⊥ is antisymmetric		u =	b⊥ • c
	-b⊥ • d					d⊥ • b

This is the formula for the parent lines, not the segments. For that, if the segments intersect at a point I, we substituting the value of t

I = A + b	(	d⊥ • c	)
		d⊥ • b

This is the intersection point.

4.6.1 The Circle through Three Points

Besides a triangle and a plane, three points designate a circle. This is the ex-circle or circumscribed circle of the triangle. The center must lie on the perpendicular bisector of each side of the triangle.
L( t ) = ½( A + B ) + ( B - A )^⊥t
Lets call the points A, B, and C, and define three vectors as:
a = B - A;
b = C - B;
c = A - C;
The perpendicular bisector of AB is the midpoint of AB and the direction perpendicular to it. The midpoint is A + a/2. The perpendicular to AB is a^⊥. The perpendicular bisector is now
A + a/2 + a^⊥t
For AC we have
A - c/2 + c^⊥u
The center lies where these two perpendicular bisectors meet
a^⊥t = b/2 + c^⊥u
where a + b + c = 0. To eliminate u we dot both sides with c
t = ( 1/2 )( b • c )/( a^⊥ • c ).
Therefore

center = A +	1	(	a + a⊥	b • c	)
	2			a⊥ • c

radius =	| a |	√	(	b • c	)	²	+ 1
	2			a&perp • c

4.7

  INTERSECTIONS OF LINES WITH PLANES; CLIPPING

Consider a ray starting at point A, with parametric description
R(t) = A + ct.
We want to know where it intersects an object with point normal form
n • ( P - B ) = 0.
The hit occurs at time t = t_hit. At this value of t the ray and the plane have the same coordinates. Solve for this time,
n • ( A + ct_hit - B ) = 0
can be rewritten as
n • ( A - B ) + n • ct_hit = 0
which is a linear equation in t_hit

thit =	n • ( B - A )
	n • c

If n • c = 0 they are parallel. Otherwise, the coordinates of the hit are
P_hit = A + ct_hit.

A2.1.2 Multiplying Two Matrices

Two matrices can be multiplied if they conform, or, the number of columns of the first equals the number of rows of the second. Then, the terms of the product are the dot product of each row of the first with each column of the second.

( 2 0 6 -3 )( 8 1 -4 0 0 5 7 1

6 2 ) -1 1 3 1 -5 8

=	(	6×2 - 1×0 + 3×6 - 5×-3		2×2 + 1×0 + 1×6 + 8×-3	)
		6×8 - 1×1 + 3×-4 - 5×0		2×8 + 1×1 + 1×-4 + 8×0
		6×0 - 1×5 + 3×7 - 5×1		2×0 + 1×5 + 1×7 + 8×1

=	(	45	-14	)
		35	13
		11	20

In forming a product of two matrices A and B, the order in which the matrices are taken makes a difference. For the expression AB, we say "A premultiplies B" or "A is postmultiplied by B".

CHAPTER 5: Transformations of Objects

5.2.3 Geometric Effects of Elementary 2D Affine Transformations

Translation	Scaling
( 1 0 m13 ) 0 1 m23 0 0 1	( Sx 0 0 ) 0 Sx 0 0 0 1
Rotation	Shearing
( cos(θ) -sin(θ) 0 ) sin(θ) cos(θ) 0 0 0 1	( 1 x 0 ) y 1 0 0 0 1

5.2.4 The Inverse of an Affine Transformation

Translation	Scaling
( 1 0 -m13 ) 0 1 -m23 0 0 1	( 1 / Sx 0 0 ) 0 1 / Sx 0 0 0 1
Rotation	Shearing
( cos(θ) sin(θ) 0 ) -sin(θ) cos(θ) 0 0 0 1	( 1 -x 0 ) 0 1 0 0 0 1

5.2.5 Composing Affine Transformations

The order that multiple tranformations occur in matters. Multiplying matrices in different orders gives different results. The representation used, if point P is to be multiplied first by matrice M₁, then by matrice M₂, it will be represented as
MP = M₂ M₁ P
or
M( P ) = M₂( M₁( P ) ).
If you want to compose or concate matrices first,
M = M₄ M₃ M₂ M₁
or
M = M₄( M₃( M₂( M₁() ) ) ).

5.2.6 Examples of Composing 2D Transformations

• So far all the rotations have been around the origin. Suppose we want to rotate or pivot around point V. We have to translate V back to the origin, then rotate, then translate back.

• (	1	0	Vx	)(	cos(θ)	-sin(θ)	0	)(	1	0	-Vx	)
  	0	1	Vy		sin(θ)	cos(θ)	0		0	1	-Vy
  	0	0	1		0	0	1		0	0	1

  = (	cos(θ)	-sin(θ)	dx	)
  	sin(θ)	cos(θ)	dx
  	0	0	1

• Scaling and shearing use the same "shift-transform-unshift" sequence.
• A reflection around an axis other than x, y, or z has to be shifted to x, y, or z, transformed and unshifted. Call the axis of reflection x + β. The final matrix is:

  = (	cos(2β)	sin(2β)	0	)
  	sin(2β)	-cos(2β)	0
  	0	0	1

• In 2D, successive rotations are additive. That will not be the case in 3D.

5.2.7 Some Useful Properties of Affine Transformations

• Affine Transformations Preserve
  • Affine combinations of points
  • Lines and planes
  • Parallelism of lines and planes
  • Relative ratios
• The columns of the matrix reveal the transformed coordinate frame.
• If an affine plane is scaled, area ×= Scaling_x × Scaling_y

5.3

  3D AFFINE TRANSFORMATIONS

5.3.1 The Elementary 3D Transformations

Translation
(	1	0	0	m14	)
	0	1	0	m24
	0	0	1	m34
	0	0	0	1

Shearing
(	1	0	0	0	)
	f	1	0	0
	0	0	1	0
	0	0	0	1

Scaling
(	Sx	0	0	0	)
	0	Sx	0	0
	0	0	Sz	0
	0	0	0	1

ROTATIONS: In two dimensions you rotate about a point. In 3D you must also specify an axis. Remember, the rotation direction we are using is "left handed". Positive values of β cause a counterclockwise (CCW) rotation about an axis as one looks inward from a point on the positive axis toward the origin.

An x-roll: Rx(β)
(	1	0	0	0	)
	0	cos(β)	-sin(β)	0
	0	sin(β)	cos(β)	0
	0	0	0	1

A y-roll: Ry(β)
(	cos(β)	0	sin(β)	0	)
	0	1	0	0
	-sin(β)	0	cos(β)	0
	0	0	0	1

A z-roll: Rz(β)
(	cos(β)	-sin(β)	0	0	)
	sin(β)	cos(β)	0	0
	0	0	1	0
	0	0	0	1

Note that 12 of the terms in each matrix are the zeros and ones of the identity matrix. They occur in the row and column that correspond to the axis about which the rotation is being made.

5.3.3 Combining Rotations

• Unlike 2D transformations, when you combine rotations around different axis, the order that you do them in matters. Doing rotations in different orders gives different results. The terminology is that 3D rotations do not commute.
  M = R_z(β₃) R_y(β₂) R_x(β₁)
• Euler says we can combine transformations to make rotations around any arbitrary axis. There are two ways, first:
  1. Perform two rotations so that u becomes aligned with the x-axis.
  2. Do a z-roll through the angle β.
  3. Undo the two alignment rotations to restore u to its original direction.
     R_u(β) = R_y(-θ) R_z(φ) R_x(β) R_z(-φ) R_y(θ)
• Second, establish a 2D coordinate system in the plane of rotation with 2 orthogonal vectors. Everything is expressed as a linear combination of those vectors.

  Ru(β) =	(	c+(1-c)ux²	(1-c)uyux-suz	(1-c)uzux+suy	0	)
  		(1-c)uxuy-suz	c+(1-c)uy²	(1-c)uzuy-sux	0
  		(1-c)uxuz-suy	(1-c)uyuz+sux	c+(1-c)uz²	0
  		0	0	0	1

  where c = cos(β), s = sin(β), and (u_x, u_y, u_z) are the components of the unit vector u.

5.3.4 Summary of Properties of 3D Affine Transformations

• Affine transformations preserve affine combinations of points. If a + b = 1, then aP + bQ is a meaningful 3D point and T(aP + bQ) = aT(P) + bT(Q).
• Affine transformations preserve lines and planes. Straightness is preserved: The image T(L) of a line L in 3D space is another straight line; the image T(W) of a of a plane W in 3D space is another plane.
• Parallelism of lines and planes is preserved. If W and Z are parallel lines (or planes), then T(W) and T(Z) are also parallel.
• Relative Ratios Are Preserved. If P is a fraction ƒ of the way from point A to point B, then T(P) is the same fraction ƒ of the way from point T(A) to T(B).
• The Columns of the Matrix Reveal the Transformed Coordinate Frame. If the columns of M are the vectors m₁, m₂, and m₃ and the point m₄, the transformation maps the frame (i,j,k,γ) to the frame (m₁, m₂, m₃, m₄).
• Effect of Transformations of the Volumes of Figures. If the 3D object D has volume V, then the image T(D) of D has volume |det M|V, where |det M| is the absolute value of the determinant of M.
• Every Affine Transformation Is Composed of Elementary Operations. A 3D affine transformation may be decomposed into elementary transformations, in several ways.

5.4

  CHANGING COORDINATE SYSTEMS

Transforming Points
To apply a sequence of transformations T₁(), T₂(), T₃() (in that order) to a point P, form the matrix:
M = M₃ × M₂ × M₁.
Then P is transformed to MP. To compose each successive transformation M_i, you must premultiply by M_i.
Transforming the Coordinate System
To apply a sequence of transformations T₁(), T₂(), T₃() (in that order) to the coordinate system, form the matrix:
M = M₁ × M₂ × M₃.
Then a point P expressed in the transformed system has coordinates MP in the original system. To compose each additional transformation M_i, you must postmultiply by M_i.

5.5

  USING AFFINE TRANSFORMATIONS IN A PROGRAM

• We have already set up matricies for Window to Viewport mappings. Now we set up matrix modelview.
• The three operations that work on the modelview matrix are glScaled(), glTranslated(), and glRotated()
• These operations only compose a transformation with the current transformation. To initialize we need an identity transformation. That is glLoadIdentity().
• In program code, the operations are called in opposite order to the way they are applied. To rotate an object then translate it, the code is:
  glMatrixMode( GL_MODELVIEW );
  glLoadIdentity();
  object.init();
  glTranslated();
  glRotated();
  object.draw();

5.5.1 Saving the CT for Later Use

To save the current transformation use glPushMatrix() and glPopMatrix().

5.6

  DRAWING 3D SCENES WITH OPENGL

5.6.1 An Overview of the Viewing Process and the Graphics Pipeline.

The graphics pipeline starts with three matrices.
• The modelview matrix provides what this text calls the CT or current transformation. It has two functions. It handles the rotations, scaling, and translation of the objects current transformation. Then it handles the camera position. We will refer to the M-model and the V-view portions of the VM matrix.
• The projection matrix reduces everything to a cube, -1 to 1 in every direction. Then clipping takes place.
• The viewport matrix maps objects into the final screen viewport.

5.6.2 Some OpenGL Tools for Modeling and Viewing

Three Functions are Used to Set Modeling Transformations.
• First, glMatrixMode( GL_MODELVIEW ); establishes the modelview matrix as "current". Then it is modified with:
  1. glScaled( sx, sy, sz ); Postmultiply the current matrix, CT, by a matrix that performs a scaling by sx in x, sy in y, and sz in z. Put the result back into the current matrix.
  2. glTranslated( dx, dy, dz ); Postmultiply the current matrix, CT, by a matrix that performs a translation by dx in x, dy in y, and dz in z. Put the result back into the current matrix.
  3. glRotated( angle, u_x, u_y, u_z ); Postmultiply the current matrix, CT, by a matrix that performs a rotation through angle degrees about the axis that passes through the origin and the point (u_x, u_y, u_z). Put the result back into the current matrix. The equation used is in section 5.3.3.
Setting the Camera in OpenGL (for a Parallel Projection).
• glOrtho( left, right, bottom, top, near, far ); establishes a parallelipiped view volume that extends from left to right in x, bottom to top in y, and -near to -far in z. The function creates a projection matrix and postmultiplies the current matrix by it.
• Since this definition operates in eye coordinates, the camera's eye is at the origin. Hence, near and far are down the negative z-axis. Using a value of 2 for the near means to place the near plane at z = -2.
• The folowing code sets the projection matrix:
  glMatrixMode( GL_PROJECTION ); // make the projection matrix current
  glLoadIdentity(); // set it to the identy matrix
  glOrtho( left, right, bottom, top, near, far ); // multiply it by the new matrix
Positioning and Aiming the Camera.
• gluLookAt( eye.x, eye.y, eye.z, look.x, look.y, look.x, up.x, up.y, up.z ); creates the view matrix and postmultiplies the current matrix by it.
• Eye and look are straight forward, but it also takes an approximate upwards direction. The camera angle won't always be up and down.
• We want this to set the V (view) part of the modelview matrix VM. It is invoked before any modeling transformations are added, since subsequent tranformations will postmultiply the modelview matrix.

  glMatrixMode( GL_MODELVIEW );		// make the modelview matrix current
  glLoadIdentity();		// start with a unit matrix
  gluLookAt(	eye.x, eye.y, eye.x,	// eye position
  	look.x, look.y, look.z,	// "look at" point
  	up.x, up.y, up.z );	// approximate upwards direction

EXAMPLE: We want to look down at a scene. We set camera eye at (4, 4, 4), looking at the origin with look = (0, 1, 0). The upwards direction is (0, 1, 0). We want the view volume to have a width of 6.4 and height of 4.8 so its aspect ration is 640 / 480. We want the near 1 and far at 50.
glMatrixMode( GL_PROJECTION ); // set the view volume
glLoadIdentity();
glOrtho( -3.2, 3.2, -2.4, 2.4, 1, 50 );
glMatrixMode( GL_MODELVIEW ); // place and aim the camera
glLoadIdentity();
glLookAt( 4, 4, 4, 0, 1, 0, 0, 1, 0 );
To examine matricies, create and copy an array
GLfloat mat[ 16 ];
glGetFloatv( GL_MODELVIEW_MATRIX, mat );

5.6.3 Drawing Elementary Shapes Provided by OpenGL

• glutWireCube( GLdouble size ); // each side is of length size
• glutWireSphere( GLdouble radius, GLint nSlices, nStacks )
• glutWireTorus( GLdouble inRad, outRad, GLint nSlices, nStacks )
• glutWireTeapot( GLdouble size )
• glutSolidCube( GLdouble size )
• glutSolidSphere( GLdouble radius, GLint nSlices, nStacks )
• glutSolid etc.
• glutWireTetrahedron()
• glutWireOctahedron()
• glutWireDodecahedron()
• glutWireIcosahedron()
• glutWireCone( GLdouble basRad, height, GLint nSlices, nStacks )
• Tapered cylinders: When the topRad is 1, there is no taper. When it is 0, it is identical to a cone.
  GLUquadricObj * qobj = gluNewQuadric();
  gluQuadricDrawStyle( qobj, GLU_LINE );
  gluCylinder( qobj, basRad, topRad, height, nSlices, nStacks );
• Solid objects in 3D require shading. The object surface has properties. This is covered in Chapter 8 so the functions are only listed in this chapter.
  glMaterialfv( GL_FRONT, GL_AMBIENT, GLfloat [ 4 ] );
  glMaterialfv( GL_FRONT, GL_DIFFUSE, GLfloat [ 4 ] );
  glMaterialfv( GL_FRONT, GL_SPECULAR, GLfloat [ 4 ] );
  glMaterialfv( GL_FRONT, GL_SHININESS, GLfloat );

6.2.2 Finding the Normal Vectors

6.4.4 Building Segmented Extrusions: Tubes and Snakes

• If we stack extrusions end to end, and twist the individual extrusions, we make tubes of assorted shapes. Each extrusion is called a waist of the tube. It is fairly simple to think of the tube as wrapped around a curve or spine. As an example, lets examine the parametric representation of the curve
  C(t) = ( cos(t), sin(t), bt )
  for some constant b.
• Create a Frenet frame at each value of t_i. The vector T(t_i) that is tangent to the curve is computed. Then two vectors N(t_i) and B(t_i), which are perpendicular to T(t_i) and to each other, are computed. This supplies a matrix M with columns directly of N(t), B(t), T(t), and C(t), for each value of t.
  M_i = ( N(t_i) | B(t_i) | T(t_i) | C(t_i) ).
  • If C(t) is differerentiable it's derivative, C′(t), is tangent to the curve. If C(t) has componets C_x(t), C_y(t), C_z(t), the derivative is simply
    C′(t) = ( C′_x(t), C′_y(t), C′_z(t) ).
    We normalize this to unit length to obtain T(t).
  • If we form the cross product of C′(t) with any noncollinear vector we must obtain a vector perpendicular to T(t) and therefore perpendicular to the spine of the curve. A good choice is the second derivative, acceleration. Again, normalize to obtain a "unit binormal" vector.

    B(t) =	C′(t) × C″(t)
    	| C′(t) × C″(t) |

  • Obtain a vector perpendicular to both T(t) and B(t) by using the cross product again:
    N(t) = B(t) × T(t)
  • The values are:
    N(t) = ( - cos(t), - sin(t), 0 )
    
    

    B(t) =	( b sin(t), -b cos(t), 1 )
    	√ 1 + b²

    
    

    T(t) =	( - sin(t), cos(t), b )
    	√ 1 + b²

    
    C(t) = ( cos(t), sin(t), bt )
• If C(t) is complicated it may be awkward to form successive derivatives. It is possible to approximate them numerically by using

  C′ ≅	C(t + ε) - C(t - ε)
  	2ε

  
  

  C″ ≅	C(t - ε) - 2C(t) + C(t + ε)
  	ε²

6.4.5 "Discretely" Swept Surfaces of Revolution
If we employ pure rotations for the affine transformations and place all spine points at the origin, we are circularly sweeping a shape about an axis. This is called a surface of revolution. Here we show a profile and the resulting swept surface that approximates a martini glass.

6.5

  MESH APPROXIMATIONS TO SMOOTH OBJECTS

• So far our meshes tend to be "data intensive"--specified by listing the vertices of each face individually. Now we want to build meshes defined by formulas rather than data. We need to be able to shade them. The proper algorithm (Gouraud shading) requires us to find the proper normal vector at each vertex of each face.
• The basic approach is to "polygonalize" or "tesselate" the surface into a collection of flat faces small enough to gracefully change direction.

6.5.2 The Normal Vector to a Surface

• Parametrically the normal direction to each vertex is the vector from the origin to the surface point. This can be found using partial derivatives.

  n( u0, v0 ) =	(	∂p	×	∂p	)|
  		∂u		∂v		u=u0, v=v0

  and then can be normalized to unit-length.
• Implicitly it would be found using the gradient, ∇F, of F

  n( x0, y0, z0 ) = ∇F|x=x0, y=y0, z=z0 =	(	∂F,		∂F,		∂F	)|
  		∂x		∂y		∂z		x=x0, y=y0, z=z0

  and then normalized.

6.5.7 Surfaces of Revolutions

A surface of revolution is formed by a rotational sweep of a profile curve C around an axis. If the surface is given by:
P(u, v) = ( X(v)cos(u), X(v)sin(u), Z(v) )
where X and Z are functions, then the normal vector can be given by:

	•		•		•
n(u, v) = X(v) (	Z	(v)cos(u),	Z	(v)sin(u), -	X	(v) )

where the dot denotes the first derivative of the function.

CHAPTER 7: Three-Dimensional Viewing

7.2

  THE CAMERA REVISITED

• So far we have seen only parallel projections. The view volume is a parallepiped bounded by six walls, including a near and a far plane. OpenGL also supports perspective views of 3D scenes. It is similar to the camera used before, except its view volume has a different shape.
• The eye is at the apex of a rectangular pyramid, the view volume. The eye is looking into the pyramid. The opening is set by the view angle θ. Two planes are defined perpendicular to the axis of the pyramid, the near plane and the far plane. Where the planes intersect the pyramid, they form retangular windows. The windows have a certain aspect ratio, which can be set in OpenGL.
• The view volume is bounded by these planes. Anything outside is clipped off and points inside are projected onto the viewplane at corresponding point P'.

7.2.1 Setting the View Volume

• Recall that the shape of the camera view volume is encoded in the projection matrix of the graphics pipeline. It can be set using gluPerspective() and it's four parameters.

  glMatrixMode( GL_PROJECTION );	// make the projection matrix current
  glLoadIdentity();	// start with a unit matrix
  gluPerspective( viewAngle, aspectRatio, N, F );	// load appropriate values

• The parameter viewAngle, given in degrees, sets the angle between the top and bottom walls of the pyramid.
• The parameter aspectRatio sets the aspect ratio of any window parallel to the xy-plane.
• N is the distance from the eye to the near plane and F is the distance to the far plane. N and F values should be positive.
• i.e.: gluPerspecive( 60.0, 1.5, 0.3, 50.0 ) establishes the view volume to have a vertical opening of 60°, with a window aspect ratio of 1.5 and planes at z = -0.3 and z = -50.0.

7.2.2 Positioning and Pointing the Camera

Changing the aim and position of the camera is the same as performing rotations and translations. These are a part of the modelview matrix, not the projection matrix. Position and orientation are set exactly the same as parallel-projection. The only difference is that the camera resides in the projection matrix which determines the shape of the view volume.

glMatrixMode( GL_MODELVIEW );	// make the modelview matrix current
glLoadIdentity();	// start with a unit matrix
gluLookAt( eye.x, eye.y, eye.z, look.x, look.y, look.z, up.x, up.y, up.z );	// load appropriate values

The General Camera with Arbitrary Orientation and Position

• When we move and rotate this camera we need to describe this motion precisely to determine the resulting modelview matrix.
• We will attach an explicit coordinate system to the camera. This coordinate system has its origin at the eye and has three axes, usually called the u-, v-, and n-axes, that define its orientation. Its directions are given by vectors u, v, and n. Because by default, the camera looks down the negative z-axis, we say in general that the camera looks down the negative n-axis, in the direction -n. The direction u points off "to the right of" the camera, and the direction v points "upward".
• Position is easy to describe, but orientation is difficult. Aviation uses the terms pitch, heading, yaw, and roll. Pitch is the angle of the longitudinal axis (running from tail to nose and having directin -n) makes with the horizontal plane (is it pointed up or down). Roll is rotation about the longitudinal axis. Heading is the direction headed in. Given n, heading and pitch can be expressed as -n in spherical coordinates. The vector -n has longitude and latitude given by angles θ and φ, respectively. Heading is given by the longitude of -n and pitch is given by the latitude of -n.
• Pitch is rotation about the u-axis. Roll is rotation about the n-axis. The common term to change heading is yaw. Yawing left or right rotates about the v-axis.
• gluLookAt() is great for setting up the initial camera but it works with Cartesian coordinates, whereas orientation deals with angles and rotations. OpenGL doesn't give direct access to u, v, and n so we need to write code that does.

7.3

  BUILDING A CAMERA IN A PROGRAM