Interpolation¶

CalcpadCE ships three built-in interpolators — take() for nearest-value lookup, line() for piecewise linear interpolation, and spline() for smooth Hermite cubic-spline curves — exposed as plain functions usable anywhere in a worksheet.

The interpolation primer lays the three methods side-by-side on a single dataset, then re-uses them in a practical engineering case: deriving wind-load height factors \(k_z\) for two terrain categories from tabulated code values. The function-approximation study measures how well each scheme reproduces analytical exponential and trigonometric functions sampled on a coarse grid, giving a quick visual feel for accuracy and overshoot. The bilinear-style table lookup chains 1D splines along both axes to interpolate a tabulated \(T(i,\, j)\) surface — the canonical pattern for working with two-input design tables.

Sample positions, values and interpolation method are parametric, so each worksheet doubles as a sandbox for choosing the right method on real engineering data.

Interpolation of Functions¶

Reconstructs an exponential and a trigonometric function from coarse samples using nearest-value, linear and Hermite-spline interpolation. A side-by-side accuracy comparison that quickly reveals overshoot, kinks and where the smooth spline pays off.

Code:

'<h4>Exponential function</h4>
f(x) = e^x
'Values
f_1 = f(0)
f_2 = f(1)
f_3 = f(2)
f_4 = f(3)
f_5 = f(4)
f_6 = f(5)
f_7 = f(6)
n(x) = x + 1' - variable transformation
'Discrete -'f_c(x) = take(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Linear -'f_l(x) = line(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Spline -'f_s(x) = spline(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Plot'
$Plot{f(x) & f_c(x) & f_l(x) & f_s(x) @ x = 0 : 5}
'<h4>Trigonometric function</h4>
#deg
f(x) = sin(x)
'Values
f_1 = f(0)
f_2 = f(30)
f_3 = f(60)
f_4 = f(90)
f_5 = f(120)
f_6 = f(150)
f_7 = f(180)
n(x) = x/30 + 1' - variable transformation
'Discrete -'f_c(x) = take(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Linear -'f_l(x) = line(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Spline -'f_s(x) = spline(n(x); f_1; f_2; f_3; f_4; f_5; f_6; f_7)
'Plot'
$Plot{f(x) & f_c(x) & f_l(x) & f_s(x) @ x = 0 : 180}
f_s(180)

Interpolation¶

Compares take(), line() and spline() on a single dataset, then derives wind-load height factors \(k_z\) for two terrain categories by interpolating tabulated code values. A compact reference for picking the right scheme when working with discrete engineering tables.

Code:

'<h4>Data functions in CalcpadCE</h4>
f_1(x) = take(x; 9; 2; -1; 1; 4.5; 6; -4)' - discrete values (vector) <span style="color:CornflowerBlue;">━━━</span>
f_2(x) = line(x; 9; 2; -1; 1; 4.5; 6; -4)' - linear interpolation <span style="color:YellowGreen;">━━━</span>
f_3(x) = spline(x; 9; 2; -1; 1; 4.5; 6; -4)' - Hermite spline interpolation <span style="color:Tomato;">━━━</span>
'<!--'PlotHeight = 250','PlotWidth = 400'-->
$Plot{f_3(x) & f_2(x) & f_1(x) @ x = 1 : 7}
'<h4>Application - wind load height factors</h4>
'Height above terrain
Z(x) = spline(x; -0.01; 5; 10; 20; 40; 60; 80; 100; 150; 200; 250; 300; 350)
'Load for terrain type A
k_zA(x) = line(x; 0.75; 0.75; 1; 1.25; 1.5; 1.7; 1.85; 2; 2.25; 2.45; 2.65; 2.75; 2.75)
'Load for terrain type B
k_zB(x) = line(x; 0.5; 0.5; 0.65; 0.85; 1.1; 1.3; 1.45; 1.6; 1.9; 2.1; 2.3; 2.5; 2.75)
'Interpolation factor
k(z) = $Root{Z(x) = z @ x = 1 : 12}
'<!--'PlotHeight = 250','PlotWidth = 150'-->
'Chart
$Plot{k_zA(k(z))|z & k_zB(k(z))|z @ z = 0 : 50}

Double Interpolation¶

Bilinear-style table lookup that chains 1D Hermite splines along both axes to interpolate a tabulated \(T(i,\, j)\) surface. The canonical pattern for evaluating two-input design tables on intermediate values.

Code:

'Input parameters
x(j) = spline(j; 2; 4; 8)'- x values
y(i) = spline(i; 3; 9; 27)'- y values
'Output values
R_1(j) = spline(j; 11; 12; 13)' - first row
R_2(j) = spline(j; 21; 22; 23)' - second row
R_3(j) = spline(j; 31; 32; 39)' - third row
T(i; j) = spline(i; R_1(j); R_2(j); R_3(j))' - complete table
'Calculation of interpolation factors
k_x(x) = $Root{x(j) = x @ j = 1 : 3}
k_y(y) = $Root{y(i) = y @ i = 1 : 3}
'The interpolated function
f(x; y) = T(k_x(x); k_y(y))
'Usage
f(2; 3)
f(5; 5)
f(4; 9)
f(8; 27)
'Table
#val
'<!--'i = 0''j = 0'-->
'<table class="bordered">
'<tr><th></th>
#repeat 3
    '<th><!--'j = j + 1'-->'x(j)'</th>
#loop
'</tr>
#repeat 3
    '<tr><th><!--'i = i + 1''j = 0'-->'y(i)'</th>
    #repeat 3
        '<td><!--'j = j + 1'-->'T(i; j)'</td>
    #loop
    '</tr>
#loop
'</table>
#equ
PlotHeight = 300
PlotWidth = PlotHeight
'Plot
$Map{f(x; y) @ x = 2 : 8 & y = 3 : 27}

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