Errata.GraphicsGems

Errata to _Graphics Gems_, first edition, edited by Andrew Glassner
Academic Press 1990.  Code available online at http://www.graphicsgems.org/

compiled by Eric Haines (erich@acm.org) from author and reader contributions

version 1.33
date:  12/21/2024

-----

Errors in the text:

p. 3, bottom:  The equation "N . P + c = 0" is better expressed as
    "N . P - c = 0" in order to match Figure 1a. [also see p. 9 errata]

p. 5, V2 Perpendicular:  change "N <- (-Vx, Vy)" to "N <- (-Vy, Vx)"

p. 5, V2 Reflect:  change "N <- (-Vy, -Vx)" to "N <- (-Vx, -Vy)"

p. 6:  change P1 and P2 lines to
    P1 <- (( -b+sqrt(d) ) / 2a) * lv + lu
    P2 <- (( -b-sqrt(d) ) / 2a) * lv + lu
    i.e. the value computed is multiplied by the direction vector and the
    line's origin is added to this new vector to get the intersection point.

p. 9-10, starting at bottom:  If the equation on p. 3 is expressed as
    "N . P - c = 0", then change all "+ c" references to "- c" and "l-sub-c"
    to "- l-sub-c".

p. 10, for "if not l-normalized", the operation in the next line should divide
    q by "( l-sub-n dot l-sub-n)", not "Length( l-sub-n )".

p. 11, bottom: change the lower bound of the sum (sigma) from i=1 to i=0.

p. 16, Product Relations. In the right hand parts of the equations the cosine
    and sine functions should be applied only to the numerator, then the
    division by 2 is done. Specifically:

        sin(a)*sin(b) = cos(a-b)/2 - cos(a+b)/2
        cos(a)*cos(b) = cos(a-b)/2 + cos(a+b)/2
        sin(a)*cos(b) = sin(a+b)/2 + sin(a-b)/2

p. 105, last sentence of first paragraph:  "ajacent" to "adjacent".

p. 216, caption for figure 2:  "54" should read "45" to be consistent with
    the figure (error in two places in caption).

p. 224, the Bit Width 23 mask is listed as 0x00400000, it should be
    0x00420000.

p. 282, 5 and 7 lines from bottom:  should read
    "then PUSH(dadRx + 1, rx, pushlx, pushrx, y-dir, -dir )"
    and
    "then PUSH(lx, dadLx-1, pushlx, pushrx, y-dir, -dir )"
    i.e. the final "dir" should be "-dir".

p. 283, 3 lines from bottom:  add an "end" above the "else begin".

p. 284, 12 lines down:  move "x <- x + 1;" to after the next "end" statement
    (move it down only one line).

p. 299, bottom: P1 and P2 as shown are actually the distances along the line
    from the line's origin (l.sub.U). Change the P1 and P2 lines to

    t1 = (-b + sqrt(d)) / 2a
    t2 = (-b - sqrt(d)) / 2a

    P1 <- l.sub.U + t1 * l.sub.V
    P2 <- l.sub.U + t2 * l.sub.V

p. 365, last line:  "Kajia" to "Kajiya".

p. 375, "revlect v" to "reflect v".

p. 395, first paragraph:  change "discussed by Haines (1989)" to "discussed by
    Haines in Glassner (1989)".

p. 406, last equation: if q is negative, the same signs should be used for the
    square root terms, i.e.

        y^2 +- y sqrt(2z - p) + z +- sqrt(z^2 - r) = 0

p. 427, equation: the y/x was flipped, and the x not multiplied through. The
    right hand side should read:

                x + 1/2 x [y/x]^2 - 1/8 x [y/x]^4 + O x [y/x]^6

    [from Ben St. John, 8/11/2004]

p. 448, last sentence of second paragraph:  change "and now nearly as simple"
    to "and not nearly as simple".

p. 463, second to last line:  change "then alpha <- alpha + pi/2" to "then
    alpha <- pi - alpha".

p. 471:  change "1/det(M)" to "det(M)" in four places, e.g., multiply the inverse by the matrix's determinant instead
   of its reciprocal to obtain the adjoint. Similarly, change "ad_{ij} = inv_{ij}/det" to "ad_{ij} = inv_{ij} * det".

p. 479, Table 1: the number of Multiplies for rotation should be 16, not 12.

p. 495, equation 5:  this should have an equal sign (=) before the
    plus-or-minus (+/-).

p. 499, middle of page:  change "and i,j,K" to "and i,j,k".

p. 503, last sentence:  change "Let P' = Rot_(alpha, N) ..." to "Let P' =
    Rot_(theta, N) ...".

p. 516, last paragraph:  a reader notes an additional reference which
    predates Berger and Salmon & Slater, namely "The Viewing Transformation,"
    Technical Memo. no. 84, Alvy Ray Smith, Computer Graphics Project,
    Lucasfilm, June 24, 1983 (rev. May 4, 1984).

p. 576: (from Mika Haarahiltunen, approved by Andrew Glassner, 576-578, issue #24)
    The equation g'(0) = mL*(dx/dy) = m0 should be g'(0) = mL*dx = m0
    The equation g'(1) = mH*(dx/dy) = m1 should be g'(1) = mH*dx = m1
    After the dy has been removed from the g'(0) and g'(1) equations, dy is no longer referenced anywhere in the article and its definition dy = yH - yL can be removed.
p. 577:
    In the first group of equations, the equation d = m1 should be d = y0 like it was written on page 576.
    In the second group of equations, the equation d = y0 should be d = 0 since y0 was defined to be 0 in the preceding paragraph.
p. 578:
	There is this equation:

        f(x) = g(clamp(xL, xH, (x - xL)/(xH - xL)))

    Since the (x - xL)/(xH - xL) part is remapping x from [xL, xH] to [0, 1], and g(x) seems to be expecting input values between [0, 1], why is the remapped value clamped between [xL, xH]? Shouldn't the remapped value be clamped between [0, 1] instead? I believe that would match the situation in Figure 4, and the equation would be:

        f(x) = g(clamp(0, 1, (x - xL)/(xH - xL)))

p. 602, second paragraph:  the matrix Tij should be:
    [ 1  0  0  0  ]
    [ 0 1/2 0  0  ]
    [ 0  0 1/4 0  ]
    [ 0  0  0 1/8 ]

p. 610:  the binomial "( n-i [over] j )" should be "( n-1 [over] j )".  This
    error appears on the fifth line of the long derivation and within the
    Zi,j definition.

p. 614:  the equation "Bi,n(t) = (1-t)Bi-1,n-1 + tBi-1,n-1(t)" should be
    "Bi,n(t) = (1-t)Bi,n-1(t) + tBi-1,n-1(t)"; the first right-hand side
    term had an incorrect subscript and was missing a "(t)".

p. 809:  the author of "Approximation of Sweep Surfaces by Tensor Product
    B-Splines" is M. (not J.) Bloomenthal.  The author is correctly
    attributed in the text (page 569).

p. 814: 5th line from bottom. "Knuth 1981" should read "Knuth 1981b" and
    "Vol. 2" should read "Vol. 1".  The reference above this should be
    "Knuth 1981a".

p. 820, 9th line from bottom:  "D.P. Greenburg" should be "D.P. Greenberg".

-----

The following are errors in the book's code listings (corrected in the online
code at http://www.graphicsgems.org/). This list is not complete, as changes
from 2013 on are tracked on Github; see the online code for the most current
and correct version.


Serious errors (ones your compiler cannot or may not catch):

p. 630: Delete FLOOR and CEILING macros (they're more like truncations).
    Change ROUND macro to (i.e. add parentheses around "a"):
    #define ROUND(a)    ((a)>0 ? (int)((a)+0.5) : -(int)(0.5-(a)))
    Change SGN macro to (i.e. change positive condition result to "1"):
    #define SGN(a)        (((a)<0) ? -1 : 1)

p. 632:  procedure declarations for routines in the "2D and 2D Vector C
     Library" (next pages) are missing from "GraphicsGems.h", e.g.

        double V2SquaredLength() ;
        double V2Length() ;
        Vector2 *V2Negate() ;
        ...

p. 638, third line from bottom:  in V3Combine change last "result->y" to
    "result->z"

p. 640:  V3MulPointByMatrix() does not work.  A separate local Point3 (e.g.
    "Point3 q ;") should be used in place of "p" for assignment and then
    passed back.

p. 649, top:  add "#include <math.h>"

p. 655, top:  add the code:
        if ((px == qx) && (py == qy))
            if ((tx == px) && (ty == py)) return 2;
            else return 0;
    in order to test for the special case where the line endpoints are
    the same.

p. 656, line 23: change "return ((R->min.x * R->min.x) < Rad2);" to
    "return ((R->min.x * R->min.x + R->max.y * R->max.y) < Rad2);"
p. 656, line 25: change
    "return ((R->min.x * R->min.x + R->min.y + R->min.y) < Rad2);" to
    "return ((R->min.x * R->min.x + R->min.y * R->min.y) < Rad2);"
    (thanks to Ben Dawson)

p. 662, line 10:  add "#include <math.h>"
p. 663, line 45:  change first "+" to "="; should read
              "VnextLeft = (Vleft=VnextLeft) + 1;"
p. 665, line 1:  change "FLOOR(...)" to "(floor((double)(...))".
p. 665, line 20, and again 43 lines later:  change "x<xRmin" to "xr<xRmin"
	to make right edges antialias properly (thanks to Andrew Bainbridge).

p. 667, line 6:  change "POLY_NMAX 8" to "POLY_NMAX 10" (for triangles and
    quadrilaterals).  Six clipping planes used on convex polygons gives
    +6 potential extra vertices generated.

p. 670-673, throughout:  change "int mask" declarations to "unsigned long
    mask" declarations.  This avoids an infinite loop occuring when the
    highest bit is set.

p. 676, line 11:  add the line "up = (double *)u;"

p. 671, line 1:  add at top of page the following test (or make sure the
    Poly_vert structure has <= 32 doubles at compilation time):

    if (sizeof(Poly_vert)/sizeof(double) > 32) {
    fprintf(stderr, "Poly_vert structure too big; must be <=32 doubles\n");
    exit(1);
    }

p. 687, line 8:  Identical points cause two points to be drawn.  Between the
    first two plot() commands, add the line:

    if ( pixels_left < 0 ) return ;

p. 714, line 20:  the last "1" in "if (i + 1 < l * 1)" should be an "l"

p. 716, line 27:  missing "/" at end of comment (if not fixed, code compiles
          but is wrong)

p. 720, lines 3 and 6 from bottom:  change "m+1>>1" to "(m+1)>>1" to establish
          correct evaluation order.

p. 737, lines 19-24, from "if ((quadrant..." to "}", should read (and note
    corrected indentations on "else" statement):

                        if (coord[i] < minB[i] || coord[i] > maxB[i])
                                return (FALSE);
                } else {
                        coord[i] = candidatePlane[i];
                }

p. 742, lines 18-24 should read:

            coeffs[ 0 ] = z - u;
            coeffs[ 1 ] = q < 0 ? -v : v;
            coeffs[ 2 ] = 1;

            num = SolveQuadric(coeffs, s);

            coeffs[ 0 ]= z + u;
            coeffs[ 1 ] = q < 0 ? v : -v;
            coeffs[ 2 ] = 1;

        The sign of "q" affects the sign of coeffs[1].

p. 744, line 6 reads:

        double    coef[MAX_ORDER];

    and should read:

        double    coef[MAX_ORDER+1];

    (thanks to Chaok Seok)

p. 745, throughout:  delete references to "lx", which is set but not used

p. 748, line 22:  change "negetive" to "negative"

p. 753, line 35:  change "int n1, n2," to "int n1=0, n2=0," so that first
    fprintf() error message has defined values

p. 756, line 15:  "unsigned int *fi=&f;" to
    "unsigned int *fi = (unsigned int *) &f;" for type
    consistency, and some compilers think "=&" means "&="

p. 766, top:  add '#include "GraphicsGems.h"' and '#include <math.h>'
    line 25:  change "det = det4x4( out );" to "det = det4x4( in );"
    throughout:  change "matrix4" to "Matrix4"

p. 774, line 5:  change "theta," to "theta = 0,"

pp. 793-794:
    There seem to be errors in the ControlPolygonFlatEnough function in the
    Graphics Gems book and the repository (NearestPoint.c). This function
    is briefly described on p. 413 of the text, and appears on pages 793-794.
    I see two main problems with it.

    The idea is to find an upper bound for the error of approximating the x
    intercept of the Bezier curve by the x intercept of the line through the
    first and last control points. It is claimed on p. 413 that this error is
    bounded by half of the difference between the intercepts of the bounding
    box. I don't see why that should be true. The line joining the first and
    last control points can be on one side of the bounding box, and the actual
    curve can be near the opposite side, so the bound should be the difference
    of the bounding box intercepts, not half of it.

    Second, we come to the implementation. The values distance[i] computed in
    the first loop are not actual distances, but squares of distances. I
    realize that minimizing or maximizing the squares is equivalent to
    minimizing or maximizing the distances.  But when the code claims that
    one of the sides of the bounding box has equation
    a * x + b * y + c + max_distance_above, where max_distance_above is one of
    those squared distances, that makes no sense to me.

    I have appended my version of the function. If you apply my code to the
    cubic Bezier curve used to test NearestPoint.c,

     static Point2 bezCurve[4] = {    /  A cubic Bezier curve    /
        { 0.0, 0.0 },
        { 1.0, 2.0 },
        { 3.0, 3.0 },
        { 4.0, 2.0 },
        };

    my code computes left_intercept = -3.0 and right_intercept = 0.0, which you
    can verify by sketching a graph. The original code computes
    left_intercept = 0.0 and right_intercept = 0.9. (from James Walker,
    jw@jwwalker.com, 9/10/2008)

    Revised code:
    static int ControlPolygonFlatEnough(V, degree)
        Point2    *V;        /* Control points    */
        int     degree;        /* Degree of polynomial    */
    {
        int     i;        /* Index variable        */
        double  value;
        double  max_distance_above;
        double  max_distance_below;
        double  error;        /* Precision of root        */
        double  intercept_1,
                intercept_2,
                left_intercept,
                right_intercept;
        double  a, b, c;    /* Coefficients of implicit    */
                /* eqn for line from V[0]-V[deg]*/
        double  det, dInv;
        double  a1, b1, c1, a2, b2, c2;

        /* Derive the implicit equation for line connecting first *'
        /*  and last control points */
        a = V[0].y - V[degree].y;
        b = V[degree].x - V[0].x;
        c = V[0].x * V[degree].y - V[degree].x * V[0].y;

        max_distance_above = max_distance_below = 0.0;

        for (i = 1; i < degree; i++)
        {
            value = a * V[i].x + b * V[i].y + c;

            if (value > max_distance_above)
            {
                max_distance_above = value;
            }
            else if (value < max_distance_below)
            {
                max_distance_below = value;
            }
        }

        /*  Implicit equation for zero line */
        a1 = 0.0;
        b1 = 1.0;
        c1 = 0.0;

        /*  Implicit equation for "above" line */
        a2 = a;
        b2 = b;
        c2 = c - max_distance_above;

        det = a1 * b2 - a2 * b1;
        dInv = 1.0/det;

        intercept_1 = (b1 * c2 - b2 * c1) * dInv;

        /*  Implicit equation for "below" line */
        a2 = a;
        b2 = b;
        c2 = c - max_distance_below;

        det = a1 * b2 - a2 * b1;
        dInv = 1.0/det;

        intercept_2 = (b1 * c2 - b2 * c1) * dInv;

        /* Compute intercepts of bounding box    */
        left_intercept = MIN(intercept_1, intercept_2);
        right_intercept = MAX(intercept_1, intercept_2);

        error = right_intercept - left_intercept;

        return (error < EPSILON)? 1 : 0;
    }

p. 799, line 32: add between first "DrawBezierCurve(...);" call and "return;":
        free((void *)bezCurve);

p. 799, bottom and p. 800, line 11:  add this pair of lines between
    "DrawBezierCurve(...);" and "return;":
        free((void *)u);
        free((void *)bezCurve);
    Also, see errata note at end of this file.

p. 800, line 5:  an additional:
    free((void *)bezCurve);
    must be done before assigning bezCurve = GenerateBezier, to avoid a leak.
    Also:
    free((void *)uPrime);
    must be added to the free list in the next correction.
    (thanks to Wolfram Esser)

    line 19:  add this pair of lines before "tHatCenter = ...":
        free((void *)u);
        free((void *)bezCurve);

p. 801, 9 from the bottom: change
        det_C0_X  = C[0][0] * X[1]    - C[0][1] * X[0];
    to:
        det_C0_X  = C[0][0] * X[1]    - C[1][0] * X[0];
    Also in this code, it is better to set alpha_l and alpha_r to 0 if
    the det_C0_C1 determinant is 0, to avoiding dividing by 0 when
    C[0][0]*C[1][1] is 0 in the corrective code. Correction has been
    made to the code itself. (thanks to Steve Anderson)

p. 802, line 2: Change " < 0.0" to " < 1.0e-6" in the alpha tests.  Otherwise
    you get coincident control points that lead to divide by zero in any
    subsequent NewtonRaphsonRootFind() call. Further correction from
    Steve Anderson of scaling the alpha_l and alpha_r < 1.0e-6 tests by
    the distance between the first and last points. See code for corrections.


p. 803, line 5 from bottom: add
    if (denominator == 0.0f) return u;
    before uPrime is computed, to avoid division by zero.
    (thanks to Steve Anderson)


Syntax errors (ones your compiler or lint will catch):

p. 633-642, throughout:  replace "};" with "}" to make lint happy
p. 640, gcd():  the variable "k" is set but not used - remove it

p. 649, line 31:  change "LengthVector3" to "V3Length"
p. 650, line 1:  bad end-of comment; delete "/"

p. 651, throughout:  Can't use "const" as a variable name, as it is a reserved
             word in ANSI C.  Use "liconst" instead.

p. 657, 659:  make "nicenum" declarations match, i.e. use either "double
        nicenum()" or "static double nicenum()" for both occurrences
p. 659:  change "exp" to "expv", since "exp()" is a math library function.

p. 660, line 11:  header missing end of comment "*/"

p. 662, line 13:  change "SYBYRES" to "SUBYRES"
        line 16:  bad space after "MODRES"
        line 42:  change "XRmax" to "xRmax"
p. 665, line 15:  missing semicolon after "int area"
        line 27:  change "O" to "0" in "if (partialArea>O)"
p. 666, line 13:  change "O" to "0" in "rightMask = O;"

p. 670, top:  add to make lint happy
        static scanline();
        static incrementalize_y();
        static incrementalize_x();
        static increment();
p. 671, line 35:  missing "{" at end of "while (y<ly && y<ry)"
p. 671-2, throughout:  add "(double *)" castings to all "incrementalize*"
        calls to make lint happy
p. 676, line 8:  change "if (tu<=0. ^ tv<=0.) {" to
        "if ( (tu<=0.) ^ (tv<=0.) ) {" to avoid precedence confusion
    end:  change "void pixelproc();" to "static void pixelproc();"

p. 681, throughout:  "y0" and "y1" are Bessel functions in the math library,
        so lint complains; ignore complaint or rename

p. 684:  change "delete" to "cdelete" and "insert" to "cinsert", since these
        routine names are already used by dbm database manager

p. 687, near end:  change "+ =" to "+="

p. 696, line 8:  add '#include "GraphicsGems.h"'
p. 700, line 5:  add '#include "GraphicsGems.h"'
p. 702-706:  change "max" to "MAX", "min" to "MIN"
p. 705, line 18:  delete "*sp", as it is not used
p. 706, end:  add '#include "GraphicsGems.h"'
p. 707, in clip:  delete "*sp", as it is not used
p. 709, line 23:  missing semicolon at end of "up = (up) ? FALSE : TRUE"
p. 710, top:  change "max" to "MAX", "min" to "MIN"

p. 713, line 26:  change ":" to ";" in "char **argv:"

p. 715, top:  pseudo-code at head of file (to advance from one element to the
            next) should be commented out.
        line 14:  missing declaration "int randmasks[32];"
    throughout:  three calls to "bitwidth()" need to cast argument to
        "(unsigned int)" to make lint happy

p. 719:  add "#include <string.h>" (or strings.h)

p. 727, line 11:  remove ")" in "static double bigC,..." line
p. 728, lines 21-23:  change "cal_q_msq" to "calc_q_msq"
    lines 23,42:  change "NULL" to "(double *)NULL" to make lint happy
        line 26:  change "con_const" to "cone_const" in
              "bigC = m1sq + con_const * q1"
    last line:  add a "}" to end albers_project procedure

p. 730:  missing inclusion of GraphicsGems.h.

p. 734, line 4:  change "exit();" to "exit(1);"

p. 736, line 20:  change "O" to "0" in "for (i=O;"

p. 739, line 12:  change "else if (D > 0)" to "else", since at this point
        D must be greater than 0; makes lint happy

p. 757, line 4:  change "{" to "[" in "sqrttab{"
    line 14:  change "= &n" to "= (unsigned int *)&n"
    line 21:  change "*num & = 0x7fffff:" to "*num &= 0x7fffff;" to fit
          ANSI C, and to fix error of ":" at end of line.
    line 22:  change "| =" to "|="
    line 27:  change ":" to ";"

p. 765, line 20,22:  change "Matrix" to "Matrix4"

p. 766, line 29:  change "exit();" to "exit(1);"

p. 774, line 2:  missing ";" at end of "long *argx, *argy"

p. 775:  P, Q, and M need to be declared as externs

p. 780, line 18:  bad start of comment for "/ re-parameterization"

p. 785, line 1:  bad start-of-comment

p. 789, lines 7,25:  change both "NULL" to "(Point2 *)NULL" to make lint happy
        in ConvertToBezierForm:  "t" is not used, can be deleted
p. 791, line 24:  remove "break;" after "return 0;"; unnecessary

p. 795, ComputeXIntercept:  "T" and "Y" do not have to be computed, since
        the result "Y" is not returned.  Note that there are many
        operations in this routine that are unnecessary (e.g. "0.0 - 0.0").

p. 797-807, throughout:  change "V2ScaleII" to "V2ScaleIII" and "Bezier" to
    "BezierII" in order to avoid name collisions with the code on pages
    787-796 (i.e. the same subroutine names are used in both, but with
    different argument types, etc).  Important only if you link in both
    of these subroutine libraries.

-----

The following are typographical errors in the comments:

p. 687, line 3:  "plottted" to "plotted"

p. 701, line 26:  change "interscetion" to "intersection"

p. 723, line 1:  "tight" is not used in the strict mathematical sense
    that the sphere generated is optimal (i.e. no smaller sphere could be
    found), but rather meaning "near optimal".

p. 728, line 10:  "latitute" to "latitude"
p. 729, line 8:  "degress" to "degrees"

p. 724, line 38:  "seperated" to "separated"
p. 725, lines 7-9:  "componant" to "component"

p. 730, line 17:  "minium" to "minimum"

p. 752, line 2:  "positve" to "positive"

p. 760, line 5:  "interger" to "integer"
p. 761, line 4:  "preceed" to "precede"

p. 766, throughout (5 times):  "determinent" to "determinant"

p. 781, lines 7,23:  "demoninator" to "denominator"

-----

Addenda:  There is additional code for Kelvin Thompson's "Rendering
Anti-Aliased Lines" gem in the online distribution of the code.

-----

Concerning page 799, Philip Schneider's Bezier code:

If you are operating in a dimensional system such that the desired error in
the fitting process is a fraction (e.g., 0.01 inches) rather than a whole
number (e.g., 2.0 pixels), then the line on page 799 reading

iterationError = error * error;

should be changed to

iterationError = ERRFACTOR * error;

where ERRFACTOR is #defined to some implementation-dependent value such as
4.0.  If this is not done, then reparameterization will never occur.  The
result is not an erroneous curve, but a suboptimal one; the algorithm will
always subdivide when the initial fit is too "loose."

(from Earl Boebert, boebert@SCTC.COM)