3D Multivariable Surface z = f(x, y) & Topographic Contour Studio

Inspect saddle points, optimization valleys, and wave fields with synchronized 3D shaded mesh projection, Marching-Squares isolines, and live symbolic gradient vectors.

3D Surface z = f(x, y) Controls

Surface Function z = f(x, y)

XY Domain Span [−L, L][-4.0, 4.0]

Mesh Grid Resolution36×36

Surface Shading Palette

Show Tangent Plane & ∇f at (x₀, y₀)

x₀1.00

y₀0.80

Synchronized 3D Surface Mesh + 2D Marching-Squares Contour Map (Click Contour to Probe)

3D Perspective Mesh (Drag to Orbit) z = f(x, y)

Topographic Isolines & Gradient ∇f f(x, y) = c

20 Canonical 3D Surfaces & Optimization Landscapes

Quadric Surfaces

Hyperbolic Paraboloid Saddle Surface z = a(x² − y²) & Minimax Point

Doubly-ruled negative Gaussian curvature surface curving upward along x and downward along y with canonical saddle point at (0,0,0).

z = 0.35*(x^2 - y^2)

Wave Optics & radially Symmetric Fields

3D Radial Sinc Sombrero Surface z = A·sin(kr)/(kr)

Axisymmetric 2D Fourier transform of a circular aperture, displaying central Airy-like peak surrounded by concentric annular ripples.

z = 3.2*sinc(1.8*sqrt(x^2 + y^2))

Quadric Surfaces & Convexity

Elliptic Paraboloid Convex Bowl z = ax² + by² — Elliptical Level Curves

Strictly convex quadratic surface with positive definite Hessian H = diag(2a, 2b) and nested elliptical contour isolines.

z = 0.28*(x^2 + 1.5*y^2) - 2

Differential Geometry

Monkey Saddle Surface z = x³ − 3xy² — Degenerate Zero-Hessian Point

Real part of complex cubic Re(z³) possessing three alternating valleys and ridges where both gradient ∇f and Hessian vanish at (0,0).

z = 0.16*(x^3 - 3*x*y^2)

Non-Convex Optimization

Himmelblau’s Optimization Benchmark f(x,y) = (x²+y−11)² + (x+y²−7)²

Multi-modal optimization test landscape featuring one central local maximum and four distinct global minima equal to zero.

z = 0.025*((x^2 + y - 11)^2 + (x + y^2 - 7)^2) - 2.5

Gradient Descent Benchmarks

Rosenbrock’s Banana Valley f(x,y) = (1 − x)² + b(y − x²)²

Narrow parabolic valley where finding the valley floor is easy, but converging to the global minimum at (1, 1) challenges first-order optimizers.

z = 0.015*((1 - x)^2 + 30*(y - x^2)^2) - 1.5

Electrostatics & Vector Calculus

Gaussian Dipole Field z = A·x·e^(−α(x² + y²)) — Paired Source & Sink

Antisymmetric smooth scalar potential exhibiting paired global maximum and minimum separated across the nodal line x = 0.

z = 3.5*x*exp(-0.45*(x^2 + y^2))

2D Acoustic Cavities & PDEs

2D Membrane Standing Wave Mode z = A·sin(kₓx)·cos(kᵧy)

Eigenfunction solution of the 2D Helmholtz wave equation on a rectangular drumhead with orthogonal nodal grid lines.

z = 2.2*sin(1.2*x)*cos(1.2*y)

Image Processing & Wavelets

2D Laplacian-of-Gaussian (Mexican Hat / Ricker Wavelet) Filter

Negative normalized Laplacian of a 2D Gaussian kernel used across computer vision and seismology for multi-scale edge detection.

z = 3.2*(1 - 0.45*(x^2 + y^2))*exp(-0.22*(x^2 + y^2))

Global Metaheuristics

Rastrigin Multi-Modal Optimization Landscape — Local Minima Lattice

Parabolic bowl superposed with periodic cosine lattice creating dozens of deceptive local minima trapping greedy solvers.

z = 0.18*(20 + (x^2 - 10*cos(2*pi*x*0.5)) + (y^2 - 10*cos(2*pi*y*0.5))) - 3

Conic Sections & Relativity

Circular Cone Surface z = √(x² + y²) — Constant Unit Gradient |∇z| = 1

Conical surface smooth everywhere except the apex singularity (0,0), exhibiting perfectly concentric equidistant circular contours.

z = sqrt(x^2 + y^2)

Spherical Geometry

Spherical Dome z = √(R² − x² − y²) — Constant Positive Curvature 1/R²

Upper hemisphere of radius R = 3 demonstrating normal vector alignment along position vector (x, y, z) and steep equatorial rim.

z = sqrt(max(0, 9 - x^2 - y^2))

2D Potential Theory & Electrostatics

2D Logarithmic Potential Funnel z = k·ln(x² + y² + ε) — Harmonic Green’s Function

Fundamental Green’s function of the 2D Laplace equation Δu = 0 governing line-charge electrostatics and ideal fluid sources.

z = 1.2*ln(x^2 + y^2 + 0.15)

Minimal Surfaces & Complex Phase

Phase Ramp & Riemann Branch Cut z = c·atan2(y, x)

Visualizes complex argument arg(x + iy) spiraling smoothly around the origin up to its branch-cut jump along the negative x-axis.

z = 1.4*atan2(y, x)

Phase Transitions & Symmetry Breaking

2D Quartic Symmetry-Breaking Potential z = α(x⁴ − βx² + y⁴ − βy²)

Landau free-energy surface displaying unstable central peak, four saddle passes, and four symmetric degenerate ground-state wells.

z = 0.18*(x^4 - 5*x^2 + y^4 - 5*y^2) + 2

Visual Cortex & Texture Filters

2D Gabor Oriented Filter Kernel z = e^(−αr²)·cos(kₓx + kᵧy)

Directionally selective localized wave packet achieving the Heisenberg uncertainty bound simultaneously in space and spatial frequency.

z = 2.8*exp(-0.22*(x^2 + y^2))*cos(2.2*x + 1.1*y)

Solid-State Tight-Binding Bands

2D Square-Lattice Tight-Binding Dispersion E(kₓ, kᵧ) = −2t(cos kₓ + cos kᵧ)

Electronic energy band surface of a 2D square crystal lattice showing Van Hove saddle singularities along diamond Fermi contours.

z = 1.8*(cos(1.3*x) + cos(1.3*y))

Spectroscopy & Field Profiles

Anisotropic 2D Lorentzian Resonance Peak z = A / (1 + αx² + βy²)

Heavy-tailed rational surface modeling focused laser beam intensity profiles and 2D nuclear magnetic resonance spectral peaks.

z = 4.2 / (1 + 0.45*(x^2 + 1.8*y^2))

Interferometry & Hyperbolic Fringes

Hyperbolic Interferogram Surface z = A·sin(α(x² − y²))

Produces nested hyperbolic crest-and-trough fringes seen in astigmatic optical wavefront interferometry.

z = 1.6*sin(0.5*(x^2 - y^2))

Ruled Surfaces & Kinematics

Regularized Plücker Conoid z = 2xy / (x² + y² + ε) — Double-Wave Gear Surface

Smoothly regularized conoid oscillating as sin(2θ) around every circle, illustrating directional limit dependence near the origin.

z = (2.4*x*y) / (x^2 + y^2 + 0.25)

Mathematical Reference & Frequently Asked Questions

How does the gradient vector ∇f relate to the 2D contour isolines?

At any differentiable point (x₀, y₀), the gradient vector ∇f = ⟨∂f/∂x, ∂f/∂y⟩ is strictly orthogonal to the level curve f(x,y) = c and points in the direction of steepest ascent.

How do I move the tangent plane probe across the surface?

Drag the x₀ and y₀ sliders in the left panel or click directly anywhere on the 2D topographic contour map to relocate the tangent plane instantaneously.