OBJ.spec: Added some mathematical formulas.

1 files changed, 51 insertions, 21 deletions

diff --git a/doc/OBJ.spec b/doc/OBJ.spec
index 13f49a5..2962721 100644
--- a/doc/OBJ.spec
+++ b/doc/OBJ.spec
@@ -2476,46 +2476,58 @@ Rational and non-rational curves and surfaces
 
 In general, any non-rational curve segment may be written as:
 
+
+    C(t) = Σ[i=0..K] d_i N_{i,n}(t)
+
 where
 
-K + 1    is the number of control points
+K + 1       is the number of control points
 
-di       are the control points
+d_i         are the control points
 
-n        is the degree of the curve
+n           is the degree of the curve
 
-Ni,n(t)          are the degree n basis functions
+N_{i,n}(t)  are the degree n basis functions
 
 Extending this to the bivariate case, any non-rational surface patch
 may be written as:
 
+    S(u,v) = Σ[i=0..K₁] Σ[j=0..K₂] d_{i,j} N_{i,m}(u) N_{j,n}(v)
+
 where:
 
-K1 + 1   is the number of control points in the u direction
+K₁ + 1      is the number of control points in the u direction
 
-K2 + 1   is the number of control points in the v direction
+K₂ + 1      is the number of control points in the v direction
 
-di,j     are the control points
+d_{i,j}     are the control points
 
-m        is the degree of the surface in the u direction
+m           is the degree of the surface in the u direction
 
-n        is the degree of the surface in the v direction
+n           is the degree of the surface in the v direction
 
-Ni,m(u)          are the degree m basis functions in the u direction
+N_{i,m}(u)  are the degree m basis functions in the u direction
 
-Nj,n(v)          are the degree n basis functions in the v direction
+N_{j,n}(v)  are the degree n basis functions in the v direction
 
 NOTE: The front of the surface is defined as the side where the u
 parameter increases to the right and the v parameter increases upward.
 
 We may extend this curve to the rational case as:
 
+       Σ[i=0..K] w_i d_i N_{i,n}(t)
+C(t) = ────────────────────────────
+         Σ[i=0..K] w_i N_{i,n}(t)
 
 
-where wi are the weights associated with the control points di.
+where w_i are the weights associated with the control points d_i.
 Similarly, a rational surface may be expressed as:
 
-where wi,j  are the weights associated with the control points di,j.
+         Σ[i=0..K₁] Σ[j=0..K₂] w_{i,j} d_{i,j} N_{i,m}(u) N_{j,n}(v)
+S(u,v) = ───────────────────────────────────────────────────────────
+             Σ[i=0..K₁] Σ[j=0..K₂] w_{i,j} N_{i,m}(u) N_{j,n}(v)
+
+where w_{i,j} are the weights associated with the control points d_{i,j}.
 
 NOTE: If a curve or surface in an .obj file is rational, it must use
 the rat option with the cstype statement and it requires some weight
@@ -2531,7 +2543,7 @@ given.
 
 
 This default weight is only reasonable for curves and surfaces whose
-basis functions sum to 1.0, such as Bezier, Cardinal, and NURB. It does
+basis functions sum to 1.0, such as Bezier, Cardinal, and NURBS. It does
 not make sense for Taylor and may or may not make sense for a
 representation given in basis-matrix form.
 
@@ -2542,9 +2554,13 @@ entire composite curve or surface using the parameter vector given with
 the parm statement.
 
 The parameter vector for a curve is a list of p global parameter values
-{t1, . . . , tp}. If t1  t < ti+1 is a point in global parameter space,
+{τ₁, . . . , τ_p}. If τ₁ ≤ t < t_{i+1} is a point in global parameter space,
 then:
 
+       τ - τ_i
+t = ─────────────
+    τ_{i+1} - τ_i
+
 is the corresponding point in local parameter space for the ith
 polynomial segment. It is this t which is used when evaluating a given
 segment of the piecewise curve. For surfaces, this mapping from global
@@ -2571,22 +2587,30 @@ Type bspline specifies arbitrary degree non-uniform B-splines which are
 commonly referred to as NURBs in their rational form. The basis
 functions are defined by the Cox-deBoor recursion formulas as:
 
+             ⎧ 1  if x_i ≤ t < x_{i+1}
+N_{i,0}(t) = ⎨
+             ⎩ 0  otherwise
+
 and:
 
+                      N_{i,n-1}(t)                      N_{i+1,n-1}(t)
+N_{i,n}(t) = (t - x_i)─────────────  +  (x_{i+n+1} - t)───────────────────
+                      x_{i+n} - x_i                    x_{i+n+1} - x_{i+1}
+
 where, by convention, 0/0 = 0.
 
-The xi  {x0, . . . ,xq} form a set known as the knot vector which is
+The x_i ∈ {x₀, . . . ,x_q} form a set known as the knot vector which is
 given by the parm statement. It is required that
 
-1.      xi  xi + 1,
+1.      x_i ≤ x_{i+1},
 
-2.      x0 < xn + 1,
+2.      x₀ < x_{n+1},
 
-3.      xq -n -1 < xq,
+3.      x_{q-n-1} < x_q,
 
-4.      xi < xi + n for 0 < i < q - n - 1,
+4.      x_i < x_{i+n} for 0 < i < q - n - 1,
 
-5.      xn  t min < tmax  xK+ 1, where [tmin, tmax] is the parameter
+5.      x_n ≤ t_min < t_max ≤ x_{K+1}, where [t_min, t_max] is the parameter
 over which the B-spline is to be evaluated, and
 
 6.      K = q - n - 1.
@@ -2606,8 +2630,14 @@ Bezier
 Type bezier specifies arbitrary degree Bezier curves and surfaces. This
 basis function is defined as:
 
+             n
+N_{i,n}(t) = i t^i (1-t)^(n-i)
+
 where:
 
+    n
+    i = n! / i!(n-i)!
+
 When using type bezier, the number of global parameter values given
 with the parm statement must be K/n + 1, where K is the number of
 control points. For surfaces, this requirement applies independently