如何在三維坐標系中畫一個標準的球？

01-08

LaTeX TikZ

documentclass{standalone} usepackage{tikz} usetikzlibrary{calc,fadings,decorations.pathreplacing}


ewcommandpgfmathsinandcos[3]{%

  pgfmathsetmacro#1{sin(#3)}%

  pgfmathsetmacro#2{cos(#3)}%

}
ewcommandLongitudePlane[3][current plane]{%

  pgfmathsinandcossinElcosEl{#2} % elevation

  pgfmathsinandcossintcost{#3} % azimuth

  	ikzset{#1/.estyle={cm={cost,sint*sinEl,0,cosEl,(0,0)}}}

}
ewcommandLatitudePlane[3][current plane]{%

  pgfmathsinandcossinElcosEl{#2} % elevation

  pgfmathsinandcossintcost{#3} % latitude

  pgfmathsetmacroyshift{cosEl*sint}

  	ikzset{#1/.estyle={cm={cost,0,0,cost*sinEl,(0,yshift)}}} %

}
ewcommandDrawLongitudeCircle[2][1]{

  LongitudePlane{angEl}{#2}

  	ikzset{current plane/.prefix style={scale=#1}}

   % angle of "visibility"

  pgfmathsetmacroangVis{atan(sin(#2)*cos(angEl)/sin(angEl))} %

  draw[current plane] (angVis:1) arc (angVis:angVis+180:1);

  draw[current plane,dashed] (angVis-180:1) arc (angVis-180:angVis:1);

}

ewcommandDrawLatitudeCircle[2][1]{ LatitudePlane{angEl}{#2} ikzset{current plane/.prefix style={scale=#1}} pgfmathsetmacrosinVis{sin(#2)/cos(#2)*sin(angEl)/cos(angEl)} % angle of "visibility" pgfmathsetmacroangVis{asin(min(1,max(sinVis,-1)))} draw[current plane] (angVis:1) arc (angVis:-angVis-180:1); draw[current plane,dashed] (180-angVis:1) arc (180-angVis:angVis:1); } egin{document} egin{tikzpicture} % "THE GLOBE" showcase defR{5} % sphere radius defangEl{35} % elevation angle filldraw[ball color=white] (0,0) circle (R); foreach in {-80,-60,...,80} { DrawLatitudeCircle[R]{ } } foreach in {-5,-35,...,-175} { DrawLongitudeCircle[R]{ } } end{tikzpicture} end{document}

代碼節錄自Tomas M. Trzeciak 的 Stereographic and cylindrical map projections。

　　基本上所有的3D軟體都有創建球體的功能，這算是三維建模的基礎。這裡使用我寫的一個軟體:數學圖形可視化工具來創建球體。軟體中使用自定義語法的腳本代碼生成數學圖形。該軟體是開源的噢。

　　在我剛學計算機圖形學的時候,就寫過生成球面的C++程序,代碼發布在球(Sphere)圖形的生成演算法。球與圓相關,一個是三維,一個是二維,可以參考下:圓,橢圓

(1)sphere的第一種公式寫法

vertices = D1:100 D2:100


t = from 0 to (PI*2) D1

r = from 0 to 1 D2

x = 2*r*sin(t)*sqrt(1-r^2) y = 2*r*cos(t)*sqrt(1-r^2) z = 1-2*(r^2)

球的網格線:

(2)sphere的另兩種公式寫法

vertices = dimension1:36 dimension2:72 u = from 0 to (2*PI) dimension1 v = from (-PI*0.5) to (PI*0.5) dimension2 r = 10.0 x = r*cos(v)*sin(u) y = r*sin(v) z = r*cos(v)*cos(u)

和

vertices = dimension1:36 dimension2:72 u = from 0 to (2*PI) dimension1 v = from 0 to (PI) dimension2 r = 10.0 x = r*sin(v)*sin(u) y = r*cos(v) z = r*sin(v)*cos(u)

兩種公式生成的圖形是一樣的

(3)彩色球

在腳本中給rgb變數設值,就能設置頂點色.

vertices = dimension1:72 dimension2:72


u = from 0 to (2*PI) dimension1

v = from (-PI*0.5) to (PI*0.5) dimension2
x = cos(v)*sin(u)

y = sin(v)

z = cos(v)*cos(u)
a = 10.0
r = (x+1.0)/2

g = (y+1.0)/2

b = (z+1.0)/2

x = a*x y = a*y z = a*z

(4)圓弧面

將球的第二維度範圍減小,即得到圓弧面

vertices = dimension1:36 dimension2:72 u = from 0 to (2*PI) dimension1 v = from (PI*0.1) to (PI*0.5) dimension2 r = 10.0 x = r*cos(v)*sin(u) y = r*sin(v) z = r*cos(v)*cos(u)

(5)橢球

#http://www.mathcurve.com/surfaces/ellipsoid/ellipsoid.shtml


vertices = D1:100 D2:100
u = from 0 to (2*PI) D1

v = from (-PI*0.5) to (PI*0.5) D2
a = rand2(1, 10)

b = rand2(1, 10)

c = rand2(1, 10)

x = a*cos(v)*sin(u) y = b*sin(v) z = c*cos(v)*cos(u)

(6)膠囊體

將球面向上下兩頭拉伸,即得到膠囊體,我也曾經寫過膠囊體的C++生成演算法膠囊體(Capsule)圖形的生成演算法

(7)刺球

將球面上頂點到球心的距離,有規律地變化,可以得到多變的球,如刺球

vertices = dimension1:129 dimension2:65


u = from 0 to (2*PI) dimension1

v = from (-PI*0.5) to (PI*0.5) dimension2
n =4
a = from 0 to 128 D1

b = from 0 to 64 D2
t = (mod(a, n) + mod(b, n))/n*4
r = 10.0 + t

x = r*cos(v)*sin(u) y = r*sin(v) z = r*cos(v)*cos(u)

照著課本繪製的球草圖：

但在標準的空間直角坐標系中球的繪製圖一直找不到。

可能真的不適合吧。