Mathematica 的几个技巧

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绘图

画曲线的交点

有多种方法。
一是利用 Mesh ，MeshFunction, 二是求出点，并使用 Graphics
eg.1:

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(*Mesh*)
 Plot[{Sin[x], 1/x},{x,0,6Pi},Mesh->{{0}}, MeshFunctions->Function[x,1/x-Sin[x]], MeshStyle->Directive[PointSize[.02], Red]]

(*Graphics*)
 pts = NSolve[Sin[x] == 1/x && 0<x<6Pi && y == 1/x,{x,y}];
Show[Plot[{Sin[x], 1/x},{x,0,6Pi}], Graphics[{Red,PointSize[0.02],Point[{x,y} /. pts]}]]

eg.2:

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(*Mesh*)
 ContourPlot[{y^2 == x^3 + 3, x^2 + y^2 == 4},{x,-5,5},{y,-5,5}, Axes->True, Mesh->{{0}}, MeshStyle->Directive[PointSize[.02], Red], MeshFunctions->Function[x, 4-x^2 - (x^3 + 3)]]

(*Graphics*)
 pts = NSolve[y^2 == x^3 + 3 && x^2 + y^2 == 4 && -5 < 0 < 5 && -5 <y < 5,{x,y}];
Show[ContourPlot[{y^2 == x^3 + 3, x^2 + y^2 == 4},{x,-5,5},{y,-5,5},Axes->True], Graphics[{Red,PointSize[0.02],Point[{x,y}/.pts]}]]

文本偏移与旋转

文本绘制使用 Text[expr,coords,offset,dir] ,其中 offset 为偏移量，单位为一个 point(一般是 1/72 inc) 。 dir 为文本的方向

eg.1 偏移

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Pt = {3,3};
Show[ {Graphics[{ Text[A,Pt,{2,2}], Text[B, Pt,{2,-2}], Text[C,Pt,{-2,2}], Text[D,Pt,{-2,-2}]}], Graphics[{Red, PointSize[.02], Point[Pt]}]}, Axes->True]

eg.2 旋转

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Graphics[{
  Text["ABC 1->0", {0, 0}, {0, 0}, {1, 0}],(* 正常 *)
  Text["ABC 1->1", {1, 0}, {0, 0}, {1, 1}],  (* 45 *)
  Text["ABC 0->1", {2, 0}, {0, 0}, {0, 1}],  (* 90 *)
  Text["ABC -1->1", {3, 0}, {0, 0}, {-1, 1}],  (* 135 *)
  Text["ABC -1->0", {4, 0}, {0, 0}, {-1, 0}], (* 180 *)
  Text["ABC 1->-1", {5, 0}, {0, 0}, {1, -1}], (* 225 *)
  Text["ABC 0->-1", {6, 0}, {0, 0}, {0, -1}] (* 270 *)
  }, Axes -> False]

计算

求模等方程

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pts =Solve[Reduce[Mod[y^2, 23] == Mod[x^3 + x + 1, 23], {x,y},Integers] && 0<= x < 23 && 0<= y  < 23];
{x,y} /. pts
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output =
{{0, 1}, {0, 22}, {1, 7}, {1, 16}, {3, 10}, {3, 13}, {4, 0}, {5,  4}, {5, 19}, {6, 4}, {6, 19}, {7, 11}, {7, 12}, {9, 7}, {9,
  16}, {11, 3}, {11, 20}, {12, 4}, {12, 19}, {13, 7}, {13, 16}, {17, 3}, {17, 20}, {18, 3}, {18, 20}, {19, 5}, {19, 18}}