數學圖形之球面,橢球面,膠囊體,刺球

從這一節開始講3D的數學圖形,爲此我準備了很久.這一節中將爲你展現如何生成球面,橢球面,膠囊體,刺球,圓弧面.

相關軟件參見:數學圖形可視化工具,使用我自定義語法的腳本代碼生成數學圖形.

在我剛學計算機圖形學的時候,就寫過生成球面的程序,代碼曾發佈在球(Sphere)圖形的生成算法.

球與圓很相關,一個是三維,一個是二維,能夠參考下:圓,橢圓

 

(1)sphere的第一種寫法

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)

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)

兩種寫法生成的圖形是同樣的

 

(3)彩色球

在腳本中給rgb變量設值,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少,即獲得圓弧面

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)橢球面

#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,即獲得膠囊體,我也曾經寫過膠囊體的生成算法膠囊體(Capsule)圖形的生成算法

 

(7)刺球

將球面上頂點到球心的距離,有規律地變化,能夠獲得多變的球,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)