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Some objects of differential geometry

Introduction

The study of the properties of continuous maps and shapes forms the foundation of differential geometry, allowing us to move from partial analytical solutions to a qualitative description of geometric structures. In this example, three canonical surfaces endowed with unique topological invariants are computationally visualized. Interactive variation of model parameters reveals the internal relationship between the local metric of objects and their global behavior in three-dimensional Euclidean space.

The hyperboloid

A hyperboloid is an open surface of the second order. It is formed by rotating a hyperbola around an axis, while a single-cavity hyperboloid is a ruled surface.

The hyperboloid equation:

In this script, we use the equations in parametric form.:

where , .

In [ ]: 
u = range(0, 2π, length=50)
v = range(-1, 1, length=50)
U = repeat(u', length(v), 1)
V = repeat(v, 1, length(u))
a, b = 1.0, 1.5
X = a .* cosh.(V) .* cos.(U)
Y = a .* cosh.(V) .* sin.(U)
Z = b .* sinh.(V)

surface(X, Y, Z;
    color = Z,
    colormap = :plasma,
    alpha = 0.9,
    colorbar = false,
    ratio = :equal,
    title = "The hyperboloid")
Out[0]:

Wave surface

A decaying wave surface is a two—dimensional scalar field that is the product of an oscillating function and a decreasing radial multiplier. The surface exhibits a localized wave packet with a maximum amplitude at the origin and its asymptotic tendency to zero at the periphery, which is typical for systems with spatial dissipation.

The equation of the surface:

Ratio — this is the attenuation parameter that determines the characteristic spatial scale of wave localization. As the value decreases, the radius of the area increases, within which the amplitude remains significant; as it increases, the waves are effectively suppressed near the center.

The square of the radial coordinate The denominator specifies an isotropic law of amplitude decay proportional to the inverse distance from the origin for large values of the argument.

In [ ]: 
u = range(-8, 8, length = 300)
v = range(-8, 8, length = 300)
k = 0.1
U, V = [u_i for u_i in u, _ in v], [v_j for _ in u, v_j in v]
X = @. U
Y = @. V
Z = @. sin(U) * cos(V) / (1 + k * (U^2 + V^2))

surface(X, Y, Z,
    fill_z = Z,                    
    linewidth = 0,               
    legend = false,
    aspect_ratio = :equal,
    colorbar = false,
    seriescolor = :hot,         
    alpha = 0.97,              
    title = "Decaying wave surface",
    xlabel = "x",
    ylabel = "y",
    zlabel = "z",
    camera = (45, 35),
    size = (800, 650))
Out[0]:

The catenoid

A catenoid is a minimal surface of rotation formed by the rotation of a chain line (catena) around the headmistress. This is the only minimal surface among all rotating surfaces having zero average curvature at each point, which makes it a physical model of a soap film stretched between two coaxial rings.

The catenoid equation in explicit form:

Catenoid equations in parametric form:

In [ ]: 
u = range(0, 2π, length=250)
v = range(-2, 2, length=150)
U, V = [u_i for u_i in u, _ in v], [v_j for _ in u, v_j in v]
X = @. cosh(V) * cos(U)
Y = @. cosh(V) * sin(U)
Z = @. V

surface(X, Y, Z,
    fill_z = Z,                  
    linewidth = 0,
    linealpha = 0,
    legend = false,
    aspect_ratio = :equal,
    colorbar = false,
    seriescolor = :cool,
    alpha = 0.85,
    title = "The catenoid",
    camera = (50, 30),
    size = (800, 650),
    fillalpha = 0.85,
    fillcolor = :cool)
Out[0]:

Conclusion

This example clearly demonstrates that even the simplest second-order parameterizations generate structurally complex and meaningful geometric images. In engineering practice, these surfaces are directly embodied: hyperboloids of rotation serve as the basis for calculating cooling towers in thermal power plants due to the exceptional rigidity of the shape, while catenoids describe the geometry of awning and cable-stayed structures that minimize material consumption. In the academic environment, such models are a necessary tool for studying minimal surfaces in the calculus of variations and the theory of singularities of differentiable maps in multidimensional spaces.