When you plot surfaces for parametric equations with Mathematica, by default, gridlines appear. So, where do these grid lines come from? Next, I will introduce the method of drawing gridlines.

computer

Mathematica

First, give a parametric equation for a surface and plot the surface: r[u_,v_]:={Sin[u],Cos[v],Sin[v] Cos[u]} This graph looks like an inflated pillow.

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In r[u,v], if v is given a definite value, then r[u,v] is a curve on the surface, called a U-curve, and the figure below is a U-curve drawn on the surface (blue line in the figure).

Use Table to draw more u curves: Table[r[u,v],{v,0,2 Pi,2 Pi/35}] Can you count the number of u curves in the graph?

Similarly, it is possible to draw a single red v curve, which is the graph for a constant value of u: r[Pi/2+1,v]

Use Table to plot several v curves: Table[r[u,v],{u,0,2 Pi,2 Pi/25}] and count the number of v curves.

When we draw the top u curve and the top v curve together, we get grid lines, but the density of the grid lines is not the same.

Use Mesh and MeshStyle directly to control the style of the grid lines and compare it with the diagram above:... 26, Mesh - > {4}, MeshStyle - > {Red, Blue},...

Each point on the grid line can be concretely represented by u and v, which are the curved coordinates of the points on the surface. It follows that a surface in three dimensions is two dimensions.