问AutoCAD如何计算只由拟合点定义的样条的端点切线？
EN

Stack Overflow用户

提问于 2020-06-19 06:22:03

回答 2查看 842关注 0票数 7

AutoCAD允许将样条实体存储在仅由拟合点定义的DXF文件中，问题是样条定义具有无限的数值正确解，Autodesk不提供从给定的拟合点计算所需参数所需的信息。

tl;dr -缺失的信息是估计的开始和结束切线的方向和大小输入切线的全局B样条插值的末端导数，谁能帮助计算这个值吗？

完整的github源代码。

我使用BricsCAD进行测试，但是"Trueview 2020“显示了相同的结果。

1.设想情况

只给出拟合点，使用不受任何约束的全局曲线插值，得到由控制顶点定义的样条：

# First spline defined by control vertices interpolated from given fit points
s = global_bspline_interpolation(points, degree=3)
msp.add_spline(dxfattribs={'color': 4, 'layer': 'Global Interpolation'}).apply_construction_tool(s)
# Second spline defined only by fit points as reference
spline = msp.add_spline(points, degree=3, dxfattribs={'layer': 'BricsCAD B-spline', 'color': 2})
doc.saveas(DIR / 'fit-points-only.dxf')

由BricsCAD从拟合点插值的样条与插值控制顶点定义的样条不匹配：

2.设想情况

除了fit点，我还在DXF文件中存储开始和结束切线值。插值由带端导数的全局曲线插值完成(Piegl & Tiller：“NURBS图书”-第9.2.2章)。

我选择了一个任意的角度(100度)作为开始和结束切线，切线大小是由“总弦长”方法估计的。

m1, m2 = estimate_end_tangent_magnitude(points, method='chord')
start_tangent = Vector.from_deg_angle(100) * m1
end_tangent = Vector.from_deg_angle(-100) * m2
# First spline defined by control vertices interpolated from given fit points and end-tangents
s = global_bspline_interpolation(points, degree=3, tangents=(start_tangent, end_tangent))
msp.add_spline(dxfattribs={'color': 4, 'layer': 'Global Interpolation'}).apply_construction_tool(s)
# Result matches the BricsCAD interpolation if fit points, start- and end
# tangents are stored explicit in the DXF file.
# Second spline defined by fit points as reference
spline = msp.add_spline(points, degree=3, dxfattribs={'layer': 'BricsCAD B-spline', 'color': 2})
# set explicit start- and end tangent as unit vectors
spline.dxf.start_tangent = Vector.from_deg_angle(100)
spline.dxf.end_tangent = Vector.from_deg_angle(-100)
doc.saveas(DIR / 'fit-points-and-tangents.dxf')

由BricsCAD插值的样条现在与插值控制顶点定义的样条完全匹配：

现在我知道插值方法是正确的，我只需要从拟合点绘制同样的样条，就像BricsCAD一样，就是从拟合点推断出的方向和大小的结束切线。

3.设想情况

我需要控制顶点来呈现B样条，但是开始切线和结束切线不像场景1中那样存储在DXF文件中。需要估计开始和结束切线，最好的方法是：“NURBS Book”、Piegl & Tiller中的"5点插值“。

tangents = estimate_tangents(points, method='5-points')
# Estimated tangent angles: (108.43494882292201, -108.43494882292201) degree
m1, m2 = estimate_end_tangent_magnitude(points, method='chord')
start_tangent = tangents[0].normalize(m1)
end_tangent = tangents[-1].normalize(m2)
# First spline defined by control vertices interpolated from given fit points and end-tangents
s = global_bspline_interpolation(points, degree=3, tangents=(start_tangent, end_tangent))
msp.add_spline(dxfattribs={'color': 4, 'layer': 'Global Interpolation'}).apply_construction_tool(s)
# Second spline defined by fit points as reference, but without explicit start- and end 
# tangents to see if my estimations are correct.
msp.add_spline(points, degree=3, dxfattribs={'layer': 'BricsCAD B-spline', 'color': 2})
doc.saveas(DIR / 'tangents-estimated.dxf')

令人惊讶的是，估计是不正确的，BricsCAD样条的切线角为101.0035408517495和-101.0035408517495度。

最烦人的是，如果我用BricsCAD角度作为输入，样条仍然不匹配，所以我假设切线幅度估计与场景2不同。

4.理论检验

从BricsCAD和样条“方法”保存的DXF文件中计算出以下值，该方法从“适合点”切换到“控制顶点”。根据这些数据，我计算了切线角和震级，tangent vector = 2nd control vertex - 1st control vertex。

required_angle = 101.0035408517495  # angle of tangent vector in degrees
required_magnitude = m1 * 1.3097943444804256  # magnitude of tangent vector
start_tangent = Vector.from_deg_angle(required_angle, required_magnitude)
end_tangent = Vector.from_deg_angle(-required_angle, required_magnitude)
s = global_bspline_interpolation(points, degree=3, tangents=(start_tangent, end_tangent))
msp.add_spline(dxfattribs={'color': 4, 'layer': 'Global Interpolation'}).apply_construction_tool(s)
msp.add_spline(points, degree=3, dxfattribs={'layer': 'BricsCAD B-spline', 'color': 2})
doc.saveas(DIR / 'theory-check.dxf')