A single chart carries a metric tensor $g(x,y)$ — a positive‑definite symmetric matrix at every point. The teal ellipses are the (scaled) infinitesimal unit balls it defines; drag the two handles to compute the geodesic between them. drag the endpoints · hover to probe the metric.
A Riemannian metric assigns to each point $p$ of the chart a positive‑definite symmetric bilinear form $g_p$ on the tangent space. In the coordinates $(x,y)$ it is the matrix $$g(x,y)=\begin{pmatrix} E & F\\ F & G\end{pmatrix},\qquad E>0,\quad EG-F^2>0,$$ and the length of an infinitesimal displacement is the line element $$ds^2 = E\,dx^2 + 2F\,dx\,dy + G\,dy^2 = \sum_{i,j} g_{ij}\,dx^i\,dx^j.$$ The norm of a tangent vector $v$ at $p$ is $\lVert v\rVert_g=\sqrt{v^\top g(p)\,v}$.
At a point $p$ the metric ball of radius $\varepsilon$ is the set of tangent vectors $$B_\varepsilon(p)=\bigl\{\,v\in T_pM : g_p(v,v)\le\varepsilon^2\,\bigr\},$$ which is an ellipse because $g_p$ is positive definite. Diagonalising $g_p = \lambda_1\,u_1u_1^\top+\lambda_2\,u_2u_2^\top$ with orthonormal eigenvectors $u_1,u_2$, the ellipse has semi‑axes $$a_i=\frac{\varepsilon}{\sqrt{\lambda_i}}\quad\text{along }u_i.$$ Where the metric is large the ball is small: a short coordinate step already costs a lot of metric length. Each ellipse is drawn centred on its grid point and enlarged by a display factor for visibility — it is a picture of the tangent space, not a region of the chart.
The length and energy of a curve $\gamma:[0,1]\to M$ are $$L[\gamma]=\int_0^1\!\sqrt{g_{\gamma}(\dot\gamma,\dot\gamma)}\,dt,\qquad E[\gamma]=\tfrac12\int_0^1\! g_{\gamma}(\dot\gamma,\dot\gamma)\,dt.$$ A geodesic is a critical point of $E$; equivalently it satisfies $$\ddot x^{\,k}+\Gamma^{k}_{ij}\,\dot x^{\,i}\dot x^{\,j}=0,\qquad \Gamma^{k}_{ij}=\tfrac12\,g^{kl}\!\left(\partial_i g_{jl}+\partial_j g_{il}-\partial_l g_{ij}\right).$$ Energy minimisers are automatically parametrised at constant speed. This page solves the boundary‑value problem $\gamma(0)=p,\ \gamma(1)=q$ directly: the curve is discretised into $N$ segments and the discrete energy $E=\tfrac{N}{2}\sum_i\Delta_i^\top g(m_i)\Delta_i$ (with midpoints $m_i$) is minimised with fixed endpoints. The stiff second‑difference part is handled implicitly through a block‑tridiagonal solve while the metric‑gradient force $\tfrac12(\partial g)(\dot\gamma,\dot\gamma)$ is applied explicitly, with a line search guaranteeing the energy decreases. For the classical charts the result is checked live against the known closed‑form geodesic (dashed).
Gaussian curvature $K$ is computed from $g$ by the Brioschi formula (numerical derivatives). The classical charts below have constant $K\in\{0,-1,+1\}$; the surface‑graph chart has varying $K$.