How to draw multiple coordinates on mathematica?

Question

For example:

How can I draw such kinds of coordinates?

Thank you~

Answer 1

You build this in Mathematica like you would do in any other descriptive language (you might want to use TikZ for this): step by step. Choosing nicer colors,adjusting the distances etc. is left as an exercise to the reader.

cosy[labels_, labelstyle_] := Flatten@{
    Arrow[{{0, 0, 0}, {1, 0, 0}}],
    Arrow[{{0, 0, 0}, {0, 1, 0}}],
    Arrow[{{0, 0, 0}, {0, 0, 1}}],
    labelstyle,
    Text[labels[[1]], {1.1, 0, 0}],
    Text[labels[[2]], {0, 1.1, 0}],
    Text[labels[[3]], {0, 0, 1.1}]
};
Graphics3D[{
    { (* Coordinate system 1 *)

   cosy[{"X", "Y", "Z"}, Darker@Orange],
        Darker@Orange,
        Text["World", {-.3, -.3, .5}]
    },
    { (* Coordinate system 2 *)

   Rotate[cosy[{"x", "y", "z"}, Blue], -30 \[Degree], {-1, 0, 1}]~
    Translate~{0, 0, -2}
    },

    { (* Connecting arrow *)
        Darker@Green,
        Arrow[{{0, 0, 0}, {0, 0, -2}}],
        Text["C(t)", {0, -.2, -1}]
    },

    { (* Red stuff *)
        Red,
        Arrow[{{0, 0, 0}, {0, 3, -1}}],
        Arrow[{{0, 0, -2}, {0, 3, -1}}],
        Text["\!\(\*SubscriptBox[\(p\), \(world\)]\)", 
    1/2 {0, 3, -1} + {0, 0, .5}],
        Text["\!\(\*SubscriptBox[\(p\), \(0\)]\)", 
    1/2 {0, 3, -1} + {0, 0, -1.5}],
        Text["p(t)", {0, 3, -1} + {0, .5, 0}]
    }
  }, Boxed -> False]

Answer 2

This is a variation on David's answer.

o = {0, 0, 0};

Clear[axis];
axis[p_, label_] := {Arrowheads[0.025], Arrow[{o, p}], Text[label, p]}

Clear[axes];
axes[labels_List, labelSize_, labelColor_] := 
 MapThread[axis, {IdentityMatrix[3], Style[#, labelSize, labelColor, Bold] & /@ labels}]

ct = 1. {0, 3, 3};
pt = {-0.5, -4, 2};

Graphics3D[{
 axes[{"\!\(\*StyleBox[\"Z\",\nFontSlant->\"Italic\"]\)", "\!\(\*StyleBox[\"X\",\nFontSlant->\"Italic\"]\)", "\!\(\*StyleBox[\"Y\",\nFontSlant->\"Italic\"]\)"}, 17, Blend[{Orange, Yellow}]], 
 Translate[Rotate[axes[{"\!\(\*StyleBox[\"z\",\nFontSlant->\"Italic\"]\)", "\!\(\*StyleBox[\"x\",\nFontSlant->\"Italic\"]\)", "\!\(\*StyleBox[\"y\",\nFontSlant->\"Italic\"]\)"}, 17, Darker@Blue], -35 Degree, {1, 1, 0}], ct], 
 Arrowheads[0.015], Darker@Green, Arrow[{{0, 0, 0}, ct}], 
 Text[Style["C(\!\(\*StyleBox[\"t\",\nFontSlant->\"Italic\"]\))", 14, Bold], Mean[{o, ct}], {-1, 1}], 
 Red, Arrow[{ct, pt}], Text[Style["\!\(\*SubscriptBox[\(p\), \(O\)]\)", 14, Bold], Mean[{ct, pt}], {-.5, -2}], 
 Arrow[{o, pt}], 
 Text[Style["\!\(\*SubscriptBox[StyleBox[\"p\",\nFontSlant->\"Italic\"], \(World\)]\)", 14, Bold], Mean[{o, pt}], {1, 0}], 
 Text[Style["\!\(\*StyleBox[\"p\",\nFontSlant->\"Italic\"]\)(\!\(\*StyleBox[\"t\",\nFontSlant->\"Italic\"]\))", 14, Bold], pt, {-.5, -1}], 
 Text[Style["\!\(\*StyleBox[\"O\",\nFontSlant->\"Italic\"]\)", 20, Bold, Darker@Blue], ct + .1 (pt - ct), {0, -5}]}, 
Boxed -> False, 
ViewVertical -> {0.37, 0.1, 2.1}, 
ViewPoint -> {1.7, 2.5, 1.6}]

The StyleBoxes appeared only because I formatted the fonts with italic and subscripts. The code looks cleaner in my notebook!

To choose values for the ViewVertical and ViewPoint options, I evaluated the expressions, then rotated the graphic to what I thought looked good, then evaluated FullOptions[] on the graphic, using values close what that output gave. Further refinement would involve tuning the angles and positions of the axes labels.