Define 1d function by two linear functions¶
Daniel Weschke
December 02, 2018
1 Objective¶
The objective is to modulate a known linear reference function \(f\)
\[f(x) = ax+b\]
with a yet unknown linear modulation \(g\)
\[g(x) = cx+d\]
to get a modulated function \(h\), which will be the product of the two linear functions.
\[\begin{split}\begin{split}
h(x) &= f(x) g(x) + e \\
&= (ax+b)(cx+d) + e \\
&= acx^2 + (ad+bc)x + bd + e
\end{split}\end{split}\]
2 Conditions¶
The first condition is that both the reference function \(f\) and the modulated function \(h\) should have the same intercept.
\[\begin{split}\begin{split}
h(0) &= f(0) \\
bd + e &= b \\
e &= b(1 - d)
\end{split}\end{split}\]
This leads to
\[\begin{split}\begin{split}
h(x) &= acx^2 + (ad+bc)x + bd + e \\
&= acx^2 + (ad+bc)x + bd + b(1 - d) \\
&= acx^2 + (ad+bc)x + b
\end{split}\end{split}\]
The second condition is that both functions should have the same integral.
\[\begin{split}\begin{split}
H &= F \\
\int\limits_0^l h(x) \dif x &= \int\limits_0^l f(x) \dif x \\
\frac{ac}{3}l^3 + \frac{ad+bc}{2}l^2 + bl &= \frac{a}{2}l^2 + bl \\
\frac{a}{2}d &= \frac{a}{2} - \left( \frac{a}{3}l + \frac{b}{2} \right)c \\
d &= 1 - \left( \frac{2}{3}l + \frac{b}{a} \right)c \\
\end{split}\end{split}\]
This leads to
\[\begin{split}\begin{split}
g(x) &= cx + 1 - \left( \frac{2}{3}l + \frac{b}{a} \right)c \\
&= c\left( x - \frac{2}{3}l - \frac{b}{a} \right) + 1 \\
&= cl\left( \frac{x}{l} - \frac{2}{3} - \frac{b}{al} \right) + 1
\end{split}\end{split}\]
and
\[\begin{split}\begin{split}
h(x) &= acx^2 + (ad+bc)x + b \\
&= acx^2 + \left(a - ac\left( \frac{2}{3}l + \frac{b}{a} \right)+bc\right)x + b \\
&= acx^2 + ax - \frac{2}{3}aclx + bcx + bcx + b \\
&= c\left(ax^2 - \frac{2}{3}alx\right) + ax + b \\
&= f^\prime l \cdot cl\left(\frac{x}{l} - \frac{2}{3}\right)\frac{x}{l} + f
\end{split}\end{split}\]
set lmargin 5
set grid
set key left
set ytics (1, '1 1/3' 4./3., '1 2/3' 5./3., 2, '2 1/3' 7./3.)
l = 1.
set xrange [0:l]
set xtics (0, '1/3' 1./3.*l, '2/3' 2./3.*l, l)
a = 1./l
b = 1.
f(x) = a*x + b
c = 1./(1.*l)
d = 1 - (2./3.*l + b/a)*c
g(x) = c*x + d
e = b*(1.-d)
h(x) = f(x)*g(x)+e
plot f(x) title 'f' ls 3, h(x) title 'h, c=l' ls 1
3 Example of usage¶
An application for this function is shown in Code_Aster Study Mirror.