How to resolve the algorithm Bernstein basis polynomials step by step in the FreeBASIC programming language
Published on 12 May 2024 09:40 PM
How to resolve the algorithm Bernstein basis polynomials step by step in the FreeBASIC programming language
Table of Contents
Problem Statement
The
n + 1
{\displaystyle n+1}
Bernstein basis polynomials of degree
n
{\displaystyle n}
are defined as Any real polynomial can written as a linear combination of such Bernstein basis polynomials. Let us call the coefficients in such linear combinations Bernstein coefficients. The goal of this task is to write subprograms for working with degree-2 and degree-3 Bernstein coefficients. A programmer is likely to have to deal with such representations. For example, OpenType fonts store glyph outline data as as either degree-2 or degree-3 Bernstein coefficients. The task is as follows: ALGOL 60 and Python implementations are provided as initial examples. The latter does the optional monomial-basis evaluations. You can use the following algorithms. They are written in unambiguous Algol 60 reference language instead of a made up pseudo-language. The ALGOL 60 example was necessary to check my work, but these reference versions are in the actual standard language designed for the printing of algorithms.
Let's start with the solution:
Step by Step solution about How to resolve the algorithm Bernstein basis polynomials step by step in the FreeBASIC programming language
Source code in the freebasic programming language
Dim Shared b0 As Double, b1 As Double, b2 As Double
Sub tobern2 (Byval a0 As Double, Byval a1 As Double, Byval a2 As Double, _
Byref b0 As Double, Byref b1 As Double, Byref b2 As Double)
' Subprogram (1): transform monomial coefficients
' a0, a1, a2 to Bernstein coefficients b0, b1, b2
b0 = a0
b1 = a0 + ((1/2) * a1)
b2 = a0 + a1 + a2
End Sub
Function evalbern2 (b0 As Double, b1 As Double, b2 As Double, t As Double) As Double
' Subprogram (2): evaluate, at t, the polynomial with
' Bernstein coefficients b0, b1, b2. Use de Casteljau's algorithm
Dim As Double s, b01, b12, b012
s = 1 - t
b01 = (s * b0) + (t * b1)
b12 = (s * b1) + (t * b2)
b012 = (s * b01) + (t * b12)
Return b012
End Function
Sub tobern3 (Byval a0 As Double, Byval a1 As Double, Byval a2 As Double, Byval a3 As Double, _
Byref b0 As Double, Byref b1 As Double, Byref b2 As Double, Byref b3 As Double)
' Subprogram (3): transform monomial coefficients
' a0, a1, a2, a3 to Bernstein coefficients b0, b1, b2, b3
b0 = a0
b1 = a0 + ((1/3) * a1)
b2 = a0 + ((2/3) * a1) + ((1/3) * a2)
b3 = a0 + a1 + a2 + a3
End Sub
Function evalbern3 (b0 As Double, b1 As Double, b2 As Double, _
b3 As Double, t As Double) As Double
' Subprogram (4): evaluate, at t, the polynomial
' with Bernstein coefficients b0, b1, b2, b3.
' Use de Casteljau's algorithm
Dim As Double s, b01, b12, b23, b012, b123, b0123
s = 1 - t
b01 = (s * b0) + (t * b1)
b12 = (s * b1) + (t * b2)
b23 = (s * b2) + (t * b3)
b012 = (s * b01) + (t * b12)
b123 = (s * b12) + (t * b23)
b0123 = (s * b012) + (t * b123)
Return b0123
End Function
Sub bern2to3 (Byval q0 As Double, Byval q1 As Double, Byval q2 As Double, _
Byref c0 As Double, Byref c1 As Double, Byref c2 As Double, Byref c3 As Double)
' Subprogram (5): transform the quadratic Bernstein
' coefficients q0, q1, q2 to the cubic Bernstein
' coefficients c0, c1, c2, c3
c0 = q0
c1 = ((1/3) * q0) + ((2/3) * q1)
c2 = ((2/3) * q1) + ((1/3) * q2)
c3 = q2
End Sub
Dim As Double p0b2, p1b2, p2b2
Dim As Double q0b2, q1b2, q2b2
Dim As Double p0b3, p1b3, p2b3, p3b3
Dim As Double q0b3, q1b3, q2b3, q3b3
Dim As Double r0b3, r1b3, r2b3, r3b3
Dim As Double pc0, pc1, pc2, pc3
Dim As Double qc0, qc1, qc2, qc3
Dim As Double x, y
Dim As Double p0m = 1, p1m = 0, p2m = 0
Dim As Double q0m = 1, q1m = 2, q2m = 3
Dim As Double r0m = 1, r1m = 2, r2m = 3, r3m = 4
tobern2 (p0m, p1m, p2m, p0b2, p1b2, p2b2)
tobern2 (q0m, q1m, q2m, q0b2, q1b2, q2b2)
Print "Subprogram (1) examples:"
Print Using " mono (#.##, #.##, #.##) --> bern (#.##, #.##, #.##)"; p0m; p1m; p2m; p0b2; p1b2; p2b2
Print Using " mono (#.##, #.##, #.##) --> bern (#.##, #.##, #.##)"; q0m; q1m; q2m; q0b2; q1b2; q2b2
Print "Subprogram (2) examples:"
x = 0.25
y = evalbern2 (p0b2, p1b2, p2b2, x)
Print Using " p (#.##) = ###.##"; x; y
x = 7.50
y = evalbern2 (p0b2, p1b2, p2b2, x)
Print Using " p (#.##) = ###.##"; x; y
x = 0.25
y = evalbern2 (q0b2, q1b2, q2b2, x)
Print Using " q (#.##) = ###.##"; x; y
x = 7.50
y = evalbern2 (q0b2, q1b2, q2b2, x)
Print Using " q (#.##) = ###.##"; x; y
tobern3 (p0m, p1m, p2m, 0, p0b3, p1b3, p2b3, p3b3)
tobern3 (q0m, q1m, q2m, 0, q0b3, q1b3, q2b3, q3b3)
tobern3 (r0m, r1m, r2m, r3m, r0b3, r1b3, r2b3, r3b3)
Print "Subprogram (3) examples:"
Print Using " mono (#.##, #.##, #.##, 0.00) --> bern (#.##, #.##, #.##, ##.##)"; p0m; p1m; p2m; p0b3; p1b3; p2b3; p3b3
Print Using " mono (#.##, #.##, #.##, 0.00) --> bern (#.##, #.##, #.##, ##.##)"; q0m; q1m; q2m; q0b3; q1b3; q2b3; q3b3
Print Using " mono (#.##, #.##, #.##, #.##) --> bern (#.##, #.##, #.##, ##.##)"; r0m; r1m; r2m; r3m; r0b3; r1b3; r2b3; r3b3
Print "Subprogram (4) examples:"
x = 0.25
y = evalbern3 (p0b3, p1b3, p2b3, p3b3, x)
Print Using " p (#.##) = ####.##"; x; y
x = 7.50
y = evalbern3 (p0b3, p1b3, p2b3, p3b3, x)
Print Using " p (#.##) = ####.##"; x; y
x = 0.25
y = evalbern3 (q0b3, q1b3, q2b3, q3b3, x)
Print Using " q (#.##) = ####.##"; x; y
x = 7.50
y = evalbern3 (q0b3, q1b3, q2b3, q3b3, x)
Print Using " q (#.##) = ####.##"; x; y
x = 0.25
y = evalbern3 (r0b3, r1b3, r2b3, r3b3, x)
Print Using " r (#.##) = ####.##"; x; y
x = 7.50
y = evalbern3 (r0b3, r1b3, r2b3, r3b3, x)
Print Using " r (#.##) = ####.##"; x; y
bern2to3 (p0b2, p1b2, p2b2, pc0, pc1, pc2, pc3)
bern2to3 (q0b2, q1b2, q2b2, qc0, qc1, qc2, qc3)
Print "Subprogram (5) examples:"
Print Using " bern (#.##, #.##, #.##) --> bern (#.##, #.##, #.##, #.##)"; p0b2; p1b2; p2b2; pc0; pc1; pc2; pc3
Print Using " bern (#.##, #.##, #.##) --> bern (#.##, #.##, #.##, #.##)"; q0b2; q1b2; q2b2; qc0; qc1; qc2; qc3
Sleep
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