[模版] 3D Grapher v1.3

Run

Tick Action: "C" Subscript, "a" "m" "e" "r" "a" "U" "p" "d" "a" "t" "e" , Baseline

every

Min Step Time: 30

Expression 146: "G" Subscript, "r" "a" "p" "h" "L" "i" "n" "e" "X" "Y" "Z" , Baseline left parenthesis, "X" Subscript, "L" , Baseline , "Y" Subscript, "L" , Baseline , "Z" Subscript, "L" , Baseline , "t" , right parenthesis equals "G" Subscript, "r" "a" "p" "h" , Baseline left parenthesis, "I" Subscript, "n" "t" "p" "l" "L" , Baseline left parenthesis, "X" Subscript, "L" , Baseline , "t" , right parenthesis , "I" Subscript, "n" "t" "p" "l" "L" , Baseline left parenthesis, "Y" Subscript, "L" , Baseline , "t" , right parenthesis , "I" Subscript, "n" "t" "p" "l" "L" , Baseline left parenthesis, "Z" Subscript, "L" , Baseline , "t" , right parenthesis , right parenthesis

146

根据给定的 X、Y、Z 列表绘制连接的线段 t：从 0 到 1

147

Expression 148: "G" Subscript, "r" "a" "p" "h" "X" "Y" "Z" "L" "i" "n" "e" , Baseline left parenthesis, "X" Subscript, "L" , Baseline , "Y" Subscript, "L" , Baseline , "Z" Subscript, "L" , Baseline , "L" Subscript, "i" , Baseline , right parenthesis equals left bracket, "G" Subscript, "r" "a" "p" "h" "L" "i" "n" "e" , Baseline left parenthesis, "X" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "x" , right bracket , "Y" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "x" , right bracket , "Z" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "x" , right bracket , "X" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "y" , right bracket , "Y" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "y" , right bracket , "Z" Subscript, "L" , Baseline left bracket, "L" Subscript, "i" , Baseline left bracket, "i" Subscript, "L" , Baseline , right bracket . "y" , right bracket , "t" , right parenthesis for "i" Subscript, "L" , Baseline equals left bracket, 1 , ... , length left parenthesis, "L" Subscript, "i" , Baseline , right parenthesis , right bracket , right bracket

148

根据给定的 X、Y、Z 列表绘制大量不连接的线段（绘制大量两个点所定义的edges） L_i: 一个点列表，每一个点代表一条线段：x代表第一个点的索引、y代表第二个点的索引

149

Expression 150: "I" Subscript, "n" "t" "p" "l" "L" , Baseline left parenthesis, "L" , "t" , right parenthesis equals "I" Subscript, "n" "t" "p" "l" , Baseline left parenthesis, "L" left bracket, 1 plus floor left parenthesis, left parenthesis, length left parenthesis, "L" , right parenthesis minus 1 , right parenthesis times "t" , right parenthesis , right bracket , "L" left bracket, mod left parenthesis, 2 plus floor left parenthesis, left parenthesis, length left parenthesis, "L" , right parenthesis minus 1 , right parenthesis times "t" , right parenthesis minus 1 , length left parenthesis, "L" , right parenthesis , right parenthesis plus 1 , right bracket , mod left parenthesis, left parenthesis, length left parenthesis, "L" , right parenthesis minus 1 , right parenthesis times "t" , 1 , right parenthesis , right parenthesis

150

为一个列表计算每个值之间的线性插值 Example: 当 L = [1,2,3]、t = 0,0.5,1 时：I_ntplL(L,t) = 1, 2, 3 t：从 0 到 1

151

Expression 152: "R" Subscript, "o" "t" "a" "t" "e" "Q" 0 , Baseline left parenthesis, "v" Subscript, "x" , Baseline , "v" Subscript, "y" , Baseline , "v" Subscript, "z" , Baseline , "A" , "x" , "y" , "z" , right parenthesis equals left bracket, negative "x" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "y" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "x" , Baseline times "v" Subscript, "y" , Baseline minus "z" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "x" , Baseline times "v" Subscript, "z" , Baseline plus "x" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared plus "z" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "y" , Baseline plus "x" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "y" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "z" , Baseline plus "x" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction plus 1 half , right parenthesis , "y" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "x" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "x" , Baseline times "v" Subscript, "y" , Baseline minus "z" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "x" , Baseline minus "y" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "z" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "y" , Baseline times "v" Subscript, "z" , Baseline plus "y" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared plus "x" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "z" , Baseline plus "y" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction plus 1 half , right parenthesis , "z" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "x" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "x" , Baseline times "v" Subscript, "z" , Baseline plus "y" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "x" , Baseline plus "z" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared minus "y" times left parenthesis, cos left parenthesis, "A" , right parenthesis minus 1 , right parenthesis times "v" Subscript, "y" , Baseline times "v" Subscript, "z" , Baseline minus "x" times sin left parenthesis, "A" , right parenthesis times "v" Subscript, "y" , Baseline minus "z" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction minus 1 half , right parenthesis times "v" squared plus "z" times left parenthesis, StartFraction, cos left parenthesis, "A" , right parenthesis Over 2 , EndFraction plus 1 half , right parenthesis , right bracketRotateQ0vx,vy,vz,A,x,y,z=−x×cosA2−12×v2x−y×cosA−1×vx×vy−z×cosA−1×vx×vz+x×cosA2−12×v2y+z×sinA×vy+x×cosA2−12×v2z−y×sinA×vz+x×cosA2+12,y×cosA2−12×v2x−x×cosA−1×vx×vy−z×sinA×vx−y×cosA2−12×v2y−z×cosA−1×vy×vz+y×cosA2−12×v2z+x×sinA×vz+y×cosA2+12,z×cosA2−12×v2x−x×cosA−1×vx×vz+y×sinA×vx+z×cosA2−12×v2y−y×cosA−1×vy×vz−x×sinA×vy−z×cosA2−12×v2z+z×cosA2+12

152

输出四元数旋转后的 xyz 列表 V (vx,vy,vz) 是旋转轴；V 与原点的距离必须为 1。 A：旋转角度 x,y,z：将要旋转的 3D 点

153

学习四元数: https://www.3dgep.com/understanding-quaternions/ https://www.youtube.com/watch?v=d4EgbgTm0Bg (3Blue1Brown) https://youtube.com/watch?v=zjMuIxRvygQ (3Blue1Brown)

154

Expression 155: "R" Subscript, "o" "t" "a" "t" "e" "Q" , Baseline left parenthesis, "v" Subscript, "x" , Baseline , "v" Subscript, "y" , Baseline , "v" Subscript, "z" , Baseline , "A" , "x" , "y" , "z" , right parenthesis equals "R" Subscript, "o" "t" "a" "t" "e" "Q" 0 , Baseline left parenthesis, StartFraction, "v" Subscript, "x" , Baseline Over StartRoot, "v" squared plus "v" squared plus "v" squared , EndRoot , EndFraction , StartFraction, "v" Subscript, "y" , Baseline Over StartRoot, "v" squared plus "v" squared plus "v" squared , EndRoot , EndFraction , StartFraction, "v" Subscript, "z" , Baseline Over StartRoot, "v" squared plus "v" squared plus "v" squared , EndRoot , EndFraction , "A" , "x" , "y" , "z" , right parenthesis

155

输出四元数旋转后的 xyz 列表 V (vx,vy,vz) 是旋转轴；V 与原点的距离可以不为1。 A：旋转角度 x,y,z：将要旋转的 3D 点

156

Expression 157: "R" Subscript, "o" "t" "a" "t" "e" "Q" "P" , Baseline left parenthesis, "V" , "A" , "L" , right parenthesis equals "R" Subscript, "o" "t" "a" "t" "e" "Q" 0 , Baseline left parenthesis, StartFraction, "V" left bracket, 1 , right bracket Over StartRoot, "V" left bracket, 1 , right bracket squared plus "V" left bracket, 2 , right bracket squared plus "V" left bracket, 3 , right bracket squared , EndRoot , EndFraction , StartFraction, "V" left bracket, 2 , right bracket Over StartRoot, "V" left bracket, 1 , right bracket squared plus "V" left bracket, 2 , right bracket squared plus "V" left bracket, 3 , right bracket squared , EndRoot , EndFraction , StartFraction, "V" left bracket, 3 , right bracket Over StartRoot, "V" left bracket, 1 , right bracket squared plus "V" left bracket, 2 , right bracket squared plus "V" left bracket, 3 , right bracket squared , EndRoot , EndFraction , "A" , "L" left bracket, 1 , right bracket , "L" left bracket, 2 , right bracket , "L" left bracket, 3 , right bracket , right parenthesis

157

输出四元数旋转后的 xyz 列表 V：旋转轴；V 与原点的距离可以不为1。 A：旋转角度 x,y,z：将要旋转的 3D 点

158

Expression 159: "S" Subscript, "p" "h" "e" "r" "e" , Baseline left parenthesis, lambda , phi , right parenthesis equals left bracket, cos left parenthesis, lambda , right parenthesis cos left parenthesis, phi , right parenthesis , sin left parenthesis, lambda , right parenthesis cos left parenthesis, phi , right parenthesis , sin left parenthesis, phi , right parenthesis , right bracket

159

以给定的经纬度返回半径为 1 的球面上的一点 (https://bit.ly/3mubei9) λ: 经度（球面上的“左右”） ϕ: 纬度（球面的“上下”）

160

Expression 161: "D" Subscript, "i" "s" "t" "O" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , right parenthesis equals StartRoot, total left parenthesis, "P" squared , right parenthesis , EndRoot

161

计算一个3D点到原点的距离

162

Expression 163: "D" Subscript, "i" "s" "t" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals StartRoot, total left parenthesis, left parenthesis, "P" Subscript, 1 , Baseline minus "P" Subscript, 2 , Baseline , right parenthesis squared , right parenthesis , EndRoot

163

计算两点之间的距离

164

Expression 165: "D" Subscript, "o" "t" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals total left parenthesis, "P" Subscript, 1 , Baseline "P" Subscript, 2 , Baseline , right parenthesis

165

计算两个 3D 点的点积

166

Expression 167: "D" Subscript, "o" "t" 2 , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals "P" Subscript, 1 , Baseline . "x" times "P" Subscript, 2 , Baseline . "x" plus "P" Subscript, 1 , Baseline . "y" times "P" Subscript, 2 , Baseline . "y"

167

计算两个 2D 点的点积

168

Expression 169: theta Subscript, "d" "i" "f" "f" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals cosine to the negative 1st power left parenthesis, round left parenthesis, StartFraction, "D" Subscript, "o" "t" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis Over "D" Subscript, "i" "s" "t" "O" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , right parenthesis "D" Subscript, "i" "s" "t" "O" , Baseline left parenthesis, "P" Subscript, 2 , Baseline , right parenthesis , EndFraction , 8 , right parenthesis , right parenthesis

169

使用点积计算两各3D点间的角度差 (不进行四舍五入，当商 = 1 或 -1 时，Desmos计算器的不精确可能导致商 >1 或 <-1)

170

Expression 171: "C" Subscript, "r" "o" "s" "s" 2 , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals "P" Subscript, 1 , Baseline . "x" times "P" Subscript, 2 , Baseline . "y" minus "P" Subscript, 1 , Baseline . "y" times "P" Subscript, 2 , Baseline . "x"

171

计算两个2D点的乘积

172

Expression 173: "C" Subscript, "r" "o" "s" "s" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , right parenthesis equals left bracket, "P" Subscript, 1 , Baseline left bracket, 2 , right bracket "P" Subscript, 2 , Baseline left bracket, 3 , right bracket minus "P" Subscript, 1 , Baseline left bracket, 3 , right bracket "P" Subscript, 2 , Baseline left bracket, 2 , right bracket , "P" Subscript, 1 , Baseline left bracket, 3 , right bracket "P" Subscript, 2 , Baseline left bracket, 1 , right bracket minus "P" Subscript, 1 , Baseline left bracket, 1 , right bracket "P" Subscript, 2 , Baseline left bracket, 3 , right bracket , "P" Subscript, 1 , Baseline left bracket, 1 , right bracket "P" Subscript, 2 , Baseline left bracket, 2 , right bracket minus "P" Subscript, 1 , Baseline left bracket, 2 , right bracket "P" Subscript, 2 , Baseline left bracket, 1 , right bracket , right bracket

173

计算两个3D点的乘积

174

Expression 175: "R" Subscript, "o" "t" "a" "t" "e" "N" "o" "r" "m" "a" "l" , Baseline left parenthesis, "L" , "V" Subscript, 1 , Baseline , "V" Subscript, 2 , Baseline , right parenthesis equals "R" Subscript, "o" "t" "a" "t" "e" "Q" "P" , Baseline left parenthesis, "C" Subscript, "r" "o" "s" "s" , Baseline left parenthesis, "V" Subscript, 1 , Baseline , "V" Subscript, 2 , Baseline plus left bracket, 0 , 0 , 10 to the negative 8th power , right bracket , right parenthesis , theta Subscript, "d" "i" "f" "f" , Baseline left parenthesis, "V" Subscript, 1 , Baseline , "V" Subscript, 2 , Baseline , right parenthesis , "L" , right parenthesis

175

将一个 xyz 列表从一个法线旋转到另一个法线 L：将要旋转后的 xyz 列表 V1：L 自身的法线 V2：L 旋转到的法线 V1 和 V2 与原点的距离不需要为 1

176

Expression 177: "R" Subscript, "o" "t" "a" "t" "e" "T" "o" "N" "o" "r" "m" "a" "l" , Baseline left parenthesis, "L" , "V" , right parenthesis equals "R" Subscript, "o" "t" "a" "t" "e" "N" "o" "r" "m" "a" "l" , Baseline left parenthesis, "L" , left bracket, 0 , 0 , 1 , right bracket , "V" , right parenthesis

177

将一个 xyz 列表从默认向上的法线旋转到另一个法线 L：将要旋转后的xyz列表 V：L 旋转到的法线；V与原点的距离不需要为 1

178

Expression 179:

179

3D Grapher 实现:

180

基于图像拖拽的视角旋转

181

XY Plane

191

Axis Lines

194

Tick lists

204

Axis Labels

210

Explicit Function Graphing

214

Implicit Function Graphing

221

Parametric Curve Graphing

234

Credit: 3D Grapher v1.3 made by Markson Chen (https://www.desmos.com/calculator/nyrbcjbp1z)

239

240

"X" plus

"X" minus

"Y" plus

"Y" minus

"Z" plus

"Z" minus

-2

-1.5

-1

-0.5

0.5

1.5

-2

-1.5

-1

-0.5

0.5

1.5

-0.5

0.5