# Small circles at mesh nodes

How to show the (16 X 4) mesh intersection points by a tiny red dots?

ParametricPlot[v { Cos[u], Sin[u]},  {u, 0, 2 Pi},{v,1,3}, Mesh-> {15,3} ] • Narasimham, please consider reverting your latest edit, Your edit makes the answers completely irrelevant (although all three answers do work for your original question). I suggest you post a new question for the 3D version of the problem. – kglr Aug 19 '17 at 20:33
• Thanks, got it, will do.. – Narasimham Aug 19 '17 at 20:39

ParametricPlot[
v {Cos[u], Sin[u]}, {u, 0, 2 Pi}, {v, 1, 3},
Mesh   -> {15, 3},
Epilog -> {
PointSize @ .02, Red,
Point @ Catenate @ Array[
Function[{u, v}, v {Cos[u], Sin[u]}],
{15 + 2, 3 + 2},
{{0, 2 Pi}, {1, 3}}
]
}] As pointed by Shutao TANG, Catenate is new so one can replace it with Flatten[#,1]&.

Points on positive side of x axis are doubled since it is the beginning and the end of u domain. But I left them for generality, like u -> {0, 1} cases.

Due to Catenate[] is a new function for V10, here is a solution for V9 or earlier version:

You just need to repalce ParametricPlot[] with Table[] and add the interval values Pi/8 and 1 to generate the intersection coordinates.

Show[
{ParametricPlot[v {Cos[u], Sin[u]}, {u, 0, 2 Pi}, {v, 1, 3},
Mesh -> {15, 3}],
Graphics[
{Red, PointSize[Medium],
Point /@
Table[
v {Cos[u], Sin[u]},
{u, 0, 2 Pi, Pi/8}, {v, 1, 3, 1/2}]}]}
] Use a one-parameter ParametricPlot to get the mesh points:

p2 = ParametricPlot[ # {Cos[u], Sin[u]} & /@ Range[1, 3, .5], {u, 0, 2 π},
Mesh -> {Range[0, 2 Pi, 2 π/16]}, PlotStyle -> None,
MeshStyle -> Directive[Red, PointSize[Large]], Axes -> False] Combine it with the original two-parameter plot using Show or using Epilog:

Show[ParametricPlot[v {Cos[u], Sin[u]}, {u, 0, 2 π}, {v, 1, 3}, Mesh -> {15, 3}], p2]


or

ParametricPlot[v {Cos[u], Sin[u]}, {u, 0, 2 π}, {v, 1, 3}, Mesh->{15, 3}, Epilog->p2[]]


both give Update: to see the nodes... sans the surface that spans them

Show[ParametricPlot[v {Cos[u], Sin[u]}, {u, 0, 2 \[Pi]}, {v, 1, 3},
Mesh -> {15, 3}, BoundaryStyle -> Gray, PlotStyle -> None], p2] • Is it possible to avoid repetition of parametrization with a single function definition? The aim of my question is to see the nodes... sans the surface that spans them. – Narasimham Aug 19 '17 at 20:38
• @Narasimham, pp2 gives just the nodes. – kglr Aug 19 '17 at 20:54
• Indeed. Just wanted to see small circles at the nodes generated using ParametricPlot3D as well. – Narasimham Aug 19 '17 at 21:08