The role of cosh in generating PDF mathematical charts

gitbox 2025-05-29

In the fields of mathematics and engineering, hyperbolic cosine function (cosh) has important application value. Especially when generating PDF documents containing mathematical formulas and charts, it is particularly critical to accurately express and draw cosh functions and their related graphics. This article will explore the key role of cosh functions in generating PDF mathematical charts from the perspective of PHP programming, and illustrate how to implement them through examples.

1. Introduction to cosh function

The cosh function is defined as:

\cosh x = \frac{e^x + e^{-x}}{2}

It is a hyperbolic function and is widely used to describe physical phenomena such as catenary lines, heat conduction, and wave equations.

2. Generate PDF mathematical chart background

In a PHP environment, libraries such as TCPDF, FPDF, etc. are usually used to generate PDF files. If you need to draw a mathematical function image in a PDF, you must calculate the function value, draw the curve, and then embed it into the PDF. The cosh function is often used as an example to show the mathematical drawing function because of its special curve shape.

3. The key role of cosh functions in mathematical chart generation

Mathematical curve display : The cosh curve is smooth and symmetrical, suitable for displaying hyperbolic function features.
Basics of data calculation : When drawing a chart, the program generates a data sequence by calculating the numerical points of cosh(x).
Combining functions and graphics : implementing curve drawing requires accurate numerical calculations, which poses a challenge to the program's mathematical library and graphical drawing interface.
Education and scientific research value : The PDF display of cosh in mathematical charts can clearly express mathematical theories and help understand the properties of hyperbolic functions.

4. PHP example: Drawing cosh function curves using TCPDF

The following example shows how to draw an image of a cosh function in PHP combined with the TCPDF library. For brevity, the examples show only the core drawing logic.

 <?php
require_once('gitbox.net/tcpdf/tcpdf.php');  // TCPDF Introduce examples

// create PDF document
$pdf = new TCPDF();
$pdf->AddPage();

// Set the drawing starting point and coordinate system size
$startX = 20;
$startY = 100;
$width = 160;
$height = 60;

// Draw the axis
$pdf->Line($startX, $startY, $startX + $width, $startY);  // X axis
$pdf->Line($startX, $startY - $height / 2, $startX, $startY + $height / 2);  // Y axis

// calculate cosh Curve Points
$points = [];
$scaleX = $width / 6;   // x axis缩放，cover -3 arrive 3
$scaleY = $height / 10; // y axis缩放

for ($x = -3; $x <= 3; $x += 0.1) {
    $y = (cosh($x) - 1);  // cosh(x) The minimum value is 1，Remove baseline
    $px = $startX + ($x + 3) * $scaleX;
    $py = $startY - $y * $scaleY;
    $points[] = ['x' => $px, 'y' => $py];
}

// draw cosh curve
for ($i = 1; $i < count($points); $i++) {
    $pdf->Line(
        $points[$i - 1]['x'], $points[$i - 1]['y'],
        $points[$i]['x'], $points[$i]['y']
    );
}

$pdf->SetFont('helvetica', '', 12);
$pdf->Text(20, 20, 'cosh functioncurve示例');

$pdf->Output('cosh_function.pdf', 'I');

// definition cosh function
function cosh($x) {
    return (exp($x) + exp(-$x)) / 2;
}
?>

5. Conclusion

Through the above example, we can see that the core role of the cosh function in generating PDF mathematical charts is reflected not only in data calculations, but also in the accuracy of graph drawing. Mastering the application of such functions is crucial to the production of high-quality and professional mathematical documents and reports.

cosh