Generate virtual terrain from mathematical functions and visualize contour maps, gradient heat maps, and cross-section profiles in real time. Learn marching squares contour drawing and fundamentals of CAE visualization.
Horizontal cross-section profile (x direction / vertical line: section position)
Theory & Key Formulas
$$\nabla f = \left(\frac{\partial f}{\partial x},\,\frac{\partial f}{\partial y}\right)$$
Maximum slope angle: $$\theta = \arctan(|\nabla f|)$$ Contour lines are orthogonal to $\nabla f$.
What is a Topographic Map / Contour Line Simulator
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When I switch the "terrain function," the shape of the mountain changes completely. Does this represent actual terrain?
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We're creating virtual terrain with mathematical models. For example, "Peaks" is a famous MATLAB demo function with multiple peaks and valleys mixed together, and "Volcano" is a volcanic terrain with a central uplift and ring-shaped ridges around it. Actual terrain data (DEM) is essentially in the form $z = f(x, y)$, so what you learn with this simulator directly applies to real-world terrain analysis.
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When I increase the "number of contour lines," the lines get packed tightly. How do I decide which setting to use?
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The basic principle is: denser lines mean steeper slopes, sparser lines mean gentler slopes. If you increase the number too much, the lines become too crowded to read; too few, and you miss fine undulations. For hiking planning, you'd use wider intervals (fewer lines) to grasp the broad area; for detailed design of a residential development, you'd use narrower intervals (more lines). That's a common distinction in practice.
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When I turn on "Show gradient vectors," lots of white arrows appear. They seem to point perpendicular to the contour lines, right?
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Sharp observation! Gradient vectors are always orthogonal to contour lines. This follows from the mathematical property that "moving along a contour line means no change in height = zero rate of change," while "moving in the gradient direction gives the maximum rate of change." Try selecting "Saddle" and see for yourself. You'll notice the arrows extend from the valley toward the ridge, intersecting the contour lines at right angles.
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When I look at the "Gradient Heatmap" tab, the bright-colored areas match where the contour lines are dense.
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Exactly! The heatmap represents the gradient magnitude ($|\nabla f|$) in color. There's a direct correspondence: bright areas = steep slopes = dense contour lines. In practical CAE, the "stress heatmap" corresponding to this is a basic output of structural analysis, where red (high stress) areas become design attention points. The intuition you gain from terrain carries directly into CAE.
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When I move the "Cross-section position" slider in the "Cross-section Profile" tab, the graph shape changes. How is this used?
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It's a "cross-section view" cutting the terrain horizontally at a fixed $y$ value. You use it to check elevation changes along a mountain ridge, or to estimate the volume of fill and cut along a design line. In CFD, the same technique is used to examine pressure or velocity changes along the center axis of a pipe. Select "Volcano" and move the cross-section slider to the center, and you'll clearly see the shape of the ring-shaped ridge in the cross-section.
Physical Model & Key Equations
When terrain height $z$ is defined as a function $f(x, y)$ over a horizontal plane, the gradient vector represents the direction and magnitude of the steepest ascent.
$$\nabla f = \left(\frac{\partial f}{\partial x},\,\frac{\partial f}{\partial y}\right)$$
$\nabla f$ : gradient vector. Its components are the partial derivatives in the $x$ and $y$ directions. It points toward the fastest increase in elevation, and its magnitude indicates slope steepness.
The maximum slope angle $\theta$ relative to the horizontal plane is calculated as follows.
A contour line is a level set of points satisfying $f(x,y) = c$. The directional derivative along a contour line is zero, so the gradient vector is always perpendicular to the contour.
$$\nabla f \cdot \mathbf{t} = 0 \quad (\mathbf{t}\text{ is tangent to the contour line})$$
This simulator draws contours with the marching squares method. Each grid cell is classified into one of $2^4=16$ cases based on whether its corners are above or below the contour level, then linear interpolation creates smooth contour segments.
Frequently Asked Questions
The Marching Squares method extracts contour lines from 2D grid data (used in this simulator). The Marching Cubes method is its 3D counterpart, used to extract isosurfaces from voxel data (e.g., CT/MRI or fluid analysis results). 3D contour displays in CAE post-processing software (like ParaView) are almost all implemented using the Marching Cubes method.
It's used everywhere in CAE. In thermal analysis, heat flux $\mathbf{q} = -k\nabla T$ (proportional to temperature gradient); in fluid analysis, pressure gradient $\nabla p$ drives flow; in structural analysis, strain is derived from the gradient (partial derivative) of displacement. The intuition you gain from terrain contour lines and their gradients applies directly to all these analyses.
A saddle point is a critical point where $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=0$, but it is neither a maximum nor a minimum. At this point, in one direction it's a peak (higher side), and in another direction it's a valley bottom (lower side). It's important as a mountain pass, a wind corridor, and a "stationary point" in optimization problems. The saddle point problem is also a key topic in machine learning loss function optimization.
It depends on the map scale and terrain relief. For general topographic maps (1/25,000), 10 m intervals (intermediate contour) and 50 m intervals (index contour) are used; for detailed design drawings, 1 m or 5 m intervals are common. In CAE contour plots, dozens of contour lines equally dividing the range of the physical quantity of interest are standard. You can intuitively experience this with the "Number of contour lines" slider in this simulator.
In Japan, the Geospatial Information Authority of Japan (GSI) provides the "Fundamental Geospatial Data Digital Elevation Model (DEM)" at 5 m mesh for free. Globally, JAXA/NASADEM (30 m mesh) and SRTM (90 m mesh) are available. By loading these with GDAL, QGIS, or the Python richdem library, you can apply the contour and gradient analysis learned in this simulator to real data.
The biggest caution is "perceptual uniformity." Rainbow color maps look flashy visually, but because the human eye over- or under-estimates certain color changes, they can easily lead to misinterpretation of data. In scientific and technical fields, perceptually uniform color maps like Viridis, Plasma, and Turbo are recommended. In this simulator, try the three types "Terrain," "Ocean," and "Heatmap" to see the difference in visual impression.
What is Topology Maps?
Topology Maps is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition โ making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Topology Maps & Contour Line Visualizer. Understanding these equations is key to interpreting the results correctly.
$\nabla f = \left(\frac{\partial f}{\partial x},\,\frac{\partial f}{\partial y}\right)$
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Topology Maps & Contour Line Visualizer are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Enter the number of contour lines (nContoursVal, range 5โ50) to define terrain resolution; more contours reveal finer topographic detail
Set the cross-section line position (sectionYNum) as a Y-coordinate value to extract a vertical slice through the generated surface
Click visualize to render the 3D terrain mesh, contour map, heat map gradient, and 2D cross-section profile simultaneously
Monitor output statistics: Max z elevation, Min z depression, Range (total relief), and Max slope in degrees to assess terrain steepness
Worked Example
For a Gaussian hill function z = 100ยทexp(โ(xยฒ + yยฒ)/50) over domain x,y โ [โ10,10] m with nContours=20: Peak elevation reaches Max z = 100 m, depression Min z = 0.5 m, giving Range = 99.5 m. Setting sectionY = 0 m reveals a symmetric profile with Max slope โ 8.2ยฐ at distance 3 m from summit. The contour lines tighten near the peak, indicating steep terrain; the heat map transitions from red (high) to blue (low), matching the radial slope gradient.
Practical Notes
Increase nContours (30โ40) for civil engineering site surveys requiring sub-meter precision on drainage paths and cut/fill boundaries
Use Max slope output to validate embankment stability; slopes exceeding 35ยฐ in clay or 45ยฐ in rock demand slope protection design per ASCE 7 geotechnical codes
Extract multiple cross-sections (vary sectionYNum across 5โ10 positions) to identify ridgelines, valleys, and saddle points for pipeline routing or foundation site assessment
Export contour data for GIS integration or numerical groundwater flow models where terrain curvature affects seepage patterns