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球坐标系中的曲线

下例为曲线在球坐标系中的表达式. 它从原点开始移动到单位球体的表面.

    
Wolfram Language code: cSpherical[t_] := {t, Pi((1 + Sin[16t + 4t ^ 2]/2)), 1 + 2t + 3t ^ 2}
Wolfram Language code: cCartesian[t_] = CoordinateTransform["Spherical" -> "Cartesian", cSpherical[t]]
单位球体内部的曲线图. 通过颜色可以注意到，开始时移动缓慢，然后在靠近表面处加速：
Wolfram Language code: Show[Graphics3D[{Opacity[.25], Sphere[]}], ParametricPlot3D[cCartesian[t], {t, 0, 1}, PlotStyle -> Thick, ColorFunction -> (Hue[.8#4]&)]]
计算曲线的长度. 大约为半径的11.2倍：
Wolfram Language code: differentialLength[t_] = Simplify@Sqrt[ cSpherical'[t]. CoordinateChartData["Spherical", "Metric", cSpherical[t]]. cSpherical'[t]]
Wolfram Language code: NIntegrate[differentialLength[t], {t, 0, 1}]