```// File: SPLINE2.BCPL
// P.Baudelaire
// December 5, 1977  4:45 PM

// The algorithms implemented by these procedures are described in
//		"Spline Curve Techniques"
//		by P.Baudelaire, R.Flegal, & R.Sproull
//		Xerox Internal Report  (May 1977)

// Uses MICROCODE floating point routines

// outgoing procedures:

external [
ParametricSpline
CubicSpline
PSerror
]

// outgoing statics:

external [
PSzone
]

static [
PSzone=0		// storage zone
]

// incoming procedures:

external [
FLD; FAD; FML; FST	// microFLOAT (Alto floating point package)
FLDI; FSB; FDV
FCM; FNEG; FSN
FPSetup

Allocate		// Alto SYSTEM
Free
Zero
]

// incoming statics:

external [
FPwork			// microFLOAT  (floating point registers)
]

// local definitions:

manifest [
naturalSpline=0
periodicSpline=1
// floating point registers: 1 to 4
ac1=1; ac2=2; ac3=3; ac4=4
// constants:
zero=5; one=6; two=7; six=8
numFPacs=9
]

structure PSVEC [
FPworkSave word
FPworkNew word
fpx word
fpy word
h word
a word
b word
c word
r word
s word
]

manifest lPSVEC= size PSVEC/16

// local statics:

static [
PSvec=0
]

let ParametricSpline(n, x, y, d1x, d2x, d3x, d1y, d2y, d3y, splineType, nil;
numargs nargs) = valof [

// default arguments, get storage, check various things
let tempVec=vec lPSVEC
if PSinit(tempVec) eq 0 resultis 0

if n ls 0 then [
n=-n
if ConvertToFP(n, lv x, lv y) eq 0 resultis 0
]

switchon nargs into [
case 9:
splineType=naturalSpline
case 10:
case 11:
if splineType ne naturalSpline &
splineType ne periodicSpline resultis PSquit(PSerror(3))
if n ls 3 then splineType=naturalSpline
endcase
default:
resultis PSquit(PSerror(4))
]

// compute parametrization (polygonal line approximation)
let h=PSallocate(lv(PSvec>>PSVEC.h), 2*n)
if h eq 0 resultis 0
for i=0 to n-2 do [
FSB(FLD(ac2, x+2*(i+1)), x+2*i)
if FSN(ac2) eq -1 then FNEG(ac2)
FSB(FLD(ac3, y+2*(i+1)), y+2*i)
if FSN(ac3) eq -1 then FNEG(ac3)
FDV((FCM(ac2, ac3) eq 1 ? ac3, ac2), two)
]

// now, compute the cubic splines x(h) & y(h)
let done=ComputeCubicSpline(n, h, x, d1x, d2x, d3x, splineType)
if done then
done=ComputeCubicSpline(n, h, y, d1y, d2y, d3y, splineType)

// reparametrize to the interval [0, 1]
if done then for i=0 to n-2 do [
FLD(ac3, FLD(ac1, h+2*i))
FST(FML(FLD(ac2, d1x+2*i), ac1), d1x+2*i)
FST(FML(FLD(ac2, d1y+2*i), ac1), d1y+2*i)
FML(ac1, ac3)
FST(FML(FLD(ac2, d2x+2*i), ac1), d2x+2*i)
FST(FML(FLD(ac2, d2y+2*i), ac1), d2y+2*i)
FML(ac1, ac3)
FST(FML(FLD(ac2, d3x+2*i), ac1), d3x+2*i)
FST(FML(FLD(ac2, d3y+2*i), ac1), d3y+2*i)
]

resultis PSquit(done)
]

and CubicSpline(n,x,y,d1y,d2y,d3y,splineType; numargs nargs) = valof [

// default arguments, get storage, check various things
let tempVec=vec lPSVEC
if PSinit(tempVec) eq 0 resultis 0

if n ls 0 then [
n=-n
if ConvertToFP(n, lv x, lv y) eq 0 resultis 0
]

switchon nargs into [
case 6:
splineType=naturalSpline
case 7:
if splineType ne naturalSpline &
splineType ne periodicSpline resultis PSquit(PSerror(3))
if n ls 3 then splineType=naturalSpline
endcase
default:
resultis PSquit(PSerror(4))
]

let h=PSallocate(lv(PSvec>>PSVEC.h), 2*n)
if h eq 0 resultis 0
for i=0 to n-2 do FST(FSB(FLD(ac1, x+2*(i+1)), x+2*i), h+2*i)

// now, compute
let done=ComputeCubicSpline(n, h, y, d1y, d2y, d3y, splineType)
if done eq 0 resultis 0
resultis PSquit()
]

and ComputeCubicSpline(n, h, y, d1, d2, d3, splineType) = valof [
// IMPORTANT: assume all arguments are right !!!!
// ONLY called by CubicSpline & ParametricSpline
let c,r,s=0,0,0
if splineType eq periodicSpline then [
if FCM(FLD(ac2,y), y+2*(n-1)) ne 0 resultis PSquit(PSerror(2))
c=PSallocate(lv(PSvec>>PSVEC.c), 2*n)
r=PSallocate(lv(PSvec>>PSVEC.r), 2*n)
s=PSallocate(lv(PSvec>>PSVEC.s), 2*n)
if (c eq 0) % (r eq 0) % (s eq 0) resultis 0
]

let a= PSallocate(lv(PSvec>>PSVEC.a), 2*n)
let b= PSallocate(lv(PSvec>>PSVEC.b), 2*n)
if (a eq 0) % (b eq 0) resultis 0

// check overlapping knots!
for i=0 to n-1 do
if FCM(zero, h+2*i) eq 0 then FST(one, h+2*i)

test splineType eq naturalSpline
ifso [
// a(0)=2*(h(0)+h(1))
// a(i)=2*(h(i)+h(i+1))-h(i)*h(i)/a(i-1)  {i=1,2,...,n-3}
for i=1 to n-3 do [
FDV(FML(FLD(ac2,h+2*i), ac2), ac1)
FLD(ac1, FST(FSB(ac4, ac2), a+2*i))
]
]

ifnot [
// a(0)=2*(h(0)+h(n-2))
// a(i)=2*(h(i)+h(i-1))-h(i-1)*h(i-1)/a(i-1)  {i=1,2,...,n-3}
for i=1 to n-3 do [
FDV(FML(FLD(ac2,h+2*(i-1)), ac2), ac1)
FLD(ac1, FST(FSB(ac4, ac2), a+2*i))
]

// c(0)=h(n-2)
FST(FLD(ac1, h+2*(n-2)), c)
// c(i)=-h(i-1)*c(i-1)/a(i-1)   {i=1,2,...,n-3}
// Notice: c(i-1) is ac1
for i=1 to n-3 do
FST(FNEG(FDV(FML(ac1, h+2*(i-1)), a+2*(i-1))), c+2*i)
]

computebc:
if n ge 3 then test splineType eq naturalSpline
ifso [
// first: D(i)=(y(i+1)-y(i))/h(i)   {i=0,1,...,n-2}
// and: b(i)=6*(D(i+1)-D(i))   {i=0,1,...,n-3}
// Notice: D(i) is ac1, D(i+1) is ac2
FDV(FSB(FLD(ac1, y+2), y), h)
for i=0 to n-3 do [
FDV(FSB(FLD(ac2, y+2*(i+2)), y+2*(i+1)), h+2*(i+1))
FST(FML(FSB(FLD(ac3, ac2), ac1), six), b+2*i)
FLD(ac1, ac2)
]
// then: b(i)=b(i)-h(i)*b(i-1)/a(i-1)  {i=1,2,...,n-3}
// Notice: b(i-1) is ac1
FLD(ac1, b)
for i=1 to n-3 do [
FDV(FML(FLD(ac2, ac1), h+2*i), a+2*(i-1))
FST(FSB(FLD(ac1, b+2*i), ac2), b+i*2)
]
]

ifnot [
// first: D(i)=(y(i+1)-y(i))/h(i)   {i=0,1,...,n-2}
// and: b(i)=6*(D(i)-D(i-1))   {i=1,2,...,n-3}
// and: b(0)= 6*(D(0)-D(n-2))
// Notice: D(i-1) is ac1, D(i) is ac2
FDV(FSB(FLD(ac1, y), y+2*(n-2)), h+2*(n-2))
for i=0 to n-3 do [
FDV(FSB(FLD(ac2, y+2*(i+1)), y+2*i), h+2*i)
FST(FML(FSB(FLD(ac3, ac2), ac1), six), b+2*i)
FLD(ac1, ac2)
]
// then: b(i)=b(i)-h(i-1)*b(i-1)/a(i-1)  {i=1,2,...,n-3}
// Notice: b(i-1) is ac1
FLD(ac1, b)
for i=1 to n-3 do [
FDV(FML(FLD(ac2, ac1), h+2*(i-1)), a+2*(i-1))
FST(FSB(FLD(ac1, b+2*i), ac2), b+i*2)
]

// r(n-2)=1  and  s(n-2)=0
FST(one,r+2*(n-2))
FST(zero,s+2*(n-2))
// r(i)=-(h(i)*r(i+1)+c(i))/a(i)   {i=n-3,...,1,0}
// s(i)=(b(i)-h(i)*s(i+1))/a(i)   {i=n-3,...,1,0}
for i=n-3 to 0 by -1 do [
FST(FDV(FNEG(ac1), a+2*i), r+2*i)
FSB(FLD(ac1, b+2*i), FML(FLD(ac2, s+2*(i+1)), h+2*i))
FST(FDV(ac1, a+2*i), s+2*i)
]
]

computed2:
// COMPUTE SECOND DERIVATIVES
test splineType eq naturalSpline
ifso [
// d2(0)=d2(n-1)=0
FST(zero,d2); FST(zero,d2+2*(n-1))
// d2(i)=(b(i-1)-h(i)*d2(i+1))/a(i-1)  {i=n-2,...,2,1}
for i=n-2 to 1 by -1 do [
FML(FLD(ac2, d2+2*(i+1)), h+2*i)
FST(FDV(FSB(FLD(ac1, b+2*(i-1)), ac2), a+2*(i-1)), d2+2*i)
]
]

ifnot [
// ac1=(6*(D(n-2)-D(n-3))-h(n-2)*s(0)-h(n-3)*s(n-3))
FDV(FSB(FLD(ac1, y+2*(n-1)), y+2*(n-2)), h+2*(n-2))
FSB(ac1, FDV(FSB(FLD(ac2, y+2*(n-2)), y+2*(n-3)), h+2*(n-3)))
FSB(FML(ac1, six), FML(FLD(ac3, s), h+2*(n-2)))
FSB(ac1, FML(FLD(ac3, s+2*(n-3)), h+2*(n-3)))

// ac2=(h(n-2)*r(0)+h(n-3)*r(n-3)+2*(h(n-3)+h(n-2)))
FAD(FML(FLD(ac2, r), h+2*(n-2)), FML(FLD(ac3, r+2*(n-3)), h+2*(n-3)))

// ac1=d2(n-2)=ac1/ac2
FST(FDV(ac1, ac2), d2+2*(n-2))

// d2(i)=r(i)*d2(n-2) + s(i)   {i=0,1,2,...,n-3}
for i=0 to n-3 do

// d2(n-1)=d2(0)
FST(FLD(ac1, d2), d2+2*(n-1))
]

computed1d3:
// COMPUTE FIRST & THIRD DERIVATIVES
// d1(i)=(y(i+1)-y(i))/h(i) - h(i)*(2*d2(i)+d2(i+1))/6
// d3(i)=(d2(i+1)-d2(i))/h(i)   {i=0,1,2,...,n-2}
for i=0 to n-2 do [
FDV(FSB(FLD(ac1, y+2*(i+1)), y+2*i), h+2*i)
FST(FSB(ac1, FDV(FML(ac2, h+2*i), six)), d1+2*i)
FST(FDV(FSB(FLD(ac1, d2+2*(i+1)), d2+2*i), h+2*i), d3+2*i)
]

resultis true
]

and ConvertToFP(n, intXloc, intYloc) = valof [
// convert coordinates from integer to floating point
let fpx=PSallocate(lv(PSvec>>PSVEC.fpx), 2*n)
let fpy=PSallocate(lv(PSvec>>PSVEC.fpy), 2*n)
if (fpx eq 0) % (fpy eq 0) resultis PSquit(PSerror(1))
for i=0 to n-1 do [
FST(FLDI(ac1, (@intXloc)!i), fpx+2*i)
FST(FLDI(ac1, (@intYloc)!i), fpy+2*i)
]
@intXloc=fpx
@intYloc=fpy
resultis true
]

and PSerror(errorCode,a1,a2,a3,a4) = valof [
(table[#77403; #1401]) ("PS.ERRORS", lv errorCode)
resultis 0
]

and PSinit(psv) = valof [
if (PSzone eq 0) resultis PSerror(0)
PSvec=psv
Zero(PSvec, lPSVEC)
// new floating point work area:
let FPworkNew= Allocate(PSzone, 4*numFPacs+1)
if FPworkNew eq 0 resultis PSerror(1)
PSvec>>PSVEC.FPworkSave=FPwork
PSvec>>PSVEC.FPworkNew=FPworkNew
FPworkNew!0=numFPacs
FPSetup(FPworkNew)
FLDI(zero,0); FLDI(one, 1); FLDI(two,2); FLDI(six,6)
resultis true
]

and PSallocate(location, m) = valof [
let b=Allocate(PSzone, m)
if b eq 0 resultis PSquit(PSerror(1))
@location=b
resultis b
]

and PSquit(result) = valof [
if PSvec eq 0 resultis result
FPSetup(PSvec>>PSVEC.FPworkSave)
PSvec>>PSVEC.FPworkSave=0
for i=0 to lPSVEC-1 do if PSvec!i ne 0 then Free(PSzone, PSvec!i)
PSvec=0
resultis result
]

```