Sketcher BSplineDecreaseKnotMultiplicity/ru: Difference between revisions

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{{Docnav
<div class="mw-translate-fuzzy">
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
{{Docnav/ru
|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]
|[[Sketcher_BSplineIncreaseKnotMultiplicity/ru|Увеличение кратности узлов]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_SwitchVirtualSpace/ru|Переключение виртуального пространства]]
|[[Sketcher_Workbench/ru|Верстак "Sketcher"]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconR=Sketcher_SwitchVirtualSpace.svg‎
|IconR=Sketcher_SwitchVirtualSpace.svg‎
|IconC=Workbench_Sketcher.svg
|IconC=Workbench_Sketcher.svg
}}
}}
</div>


<div class="mw-translate-fuzzy">
{{GuiCommand
{{GuiCommand/ru
|Name=Sketcher BSplineDecreaseKnotMultiplicity
|Name/ru=Уменьшение кратности узлов
|MenuLocation=Sketch → Sketcher B-spline tools → Decrease knot multiplicity
|Name=Sketcher_BSplineDecreaseKnotMultiplicity
|Workbenches=[[Sketcher_Workbench|Sketcher]]
|MenuLocation=Sketch → B-сплйан инструменты эскиза → Уменьшение кратности узлов
|Workbenches=[[Sketcher_Workbench/ru|Sketcher]]
|Version=0.17
|Version=0.17
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity|Sketcher Show/hide B-spline knot multiplicity]], [[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSpline Increase knot multiplicity]]
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/ru|Показать/скрыть кратность узлов B-сплайна]], [[Sketcher_BSplineIncreaseKnotMultiplicity/ru|Увеличение кратности узлов]]
}}
}}
</div>


==Description==
<span id="Description"></span>
==Описание==

Decreases the multiplicity of a B-spline knot. (See [[B-Splines|this page]] for more info about B-splines).

B-splines are basically a combination of [[B-Splines#B.C3.A9zier_curves|Bézier curves]] (nicely explained in [https://www.youtube.com/watch?v=bE1MrrqBAl8 this] and [https://www.youtube.com/watch?v=xXJylM2S72s this] video). The points where two Bézier curves are connected to form the spline are called knots. A knot on a degree ''d'' spline with the multiplicity ''m'' means that the curve left and right to the knot has at least an equal ''n'' order derivative (called ''C''<sup>''n''</sup> continuity) whereas <math>n=d-m</math>.<br/>
Here is a cubic spline (<math>d=3</math>) whose knots have the multiplicity 1. The multiplicity is indicated by the number in parentheses. The indication can be changed using the toolbar button {{Button|[[File:Sketcher_BSplineKnotMultiplicity.svg|24px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]]}}):

[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
{{Caption|B-spline where both knots have the multiplicity 1.}}

A multiplicity of 3 will change this spline so that even the first order derivatives are not equal (''C''<sup>0</sup> continuity). Here is the same spline where the left's knot multiplicity was increased to 3:


The [[Image:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|Sketcher BSplineDecreaseKnotMultiplicity]] tool decreases the multiplicity of a [[B-Splines|B-spline]] knot.
[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
{{Caption|B-spline from above with knot multiplicity 3. A control point was moved to show that the knot has ''C''<sup>0</sup> continuity.}}


<span id="Usage"></span>
A consequence of a higher multiplicity is that for the price of loosing continuity you gain local control. This means the change of one control point only affects the spline locally to this changed point. This can be seen in this example, where the spline from the first image above was taken and its second control point from the right side was moved up:
==Применение==


# Select a B-spline knot.
[[File:Sketcher_KnotMultiplicity_locality.png|400px]]
# There are several ways to invoke the tool:
{{Caption|Effect of locality due to different multiplicity.}}
#* Press the {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|Decrease knot multiplicity]]}} button.
#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineDecreaseKnotMultiplicity.svg|16px]] Decrease knot multiplicity}} option from the menu.


==Example==
One can see that the spline with knot multiplicity 1 is completely changed while the one with multiplicity 2 kept its form at its left side.


See [[Sketcher_BSplineIncreaseKnotMultiplicity#Example|Sketcher_BSplineIncreaseKnotMultiplicity]]
'''Note:''' If you decrease the multiplicity, the knot vanishes, because mathematically it appears then zero times in the knot vector, meaning there is no longer a basis function. Understanding this, requires some math, but it will also be clear when you look at the multiplicity: For example degree = 3 then multiplicity = 0 means that at the position of the knot two Bézier pieces are connected with ''C''<sup>3</sup> continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve (that is a polynom with degree 3) , this means both sides must be part of the same curve. So there is then actually no longer a knot connecting 2 different Bézier curves, the former knot is then simply a point onto one Bézier curve.


==Usage==
==Notes==


If you decrease the multiplicity of a knot to zero, the knot vanishes. Mathematically it then appears zero times in the knot vector, meaning there is no longer a basis function. Understanding this requires some math, but it will also be clear if you look at the multiplicity. For example a knot with multiplicity 0 on a B-spline with degree 3
# Select a B-spline knot, either:
means that at the position of the knot two Bézier pieces are connected with ''C<sup>3</sup>'' continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve this means that both sides must be part of the same curve. There is then effectively no longer a knot connecting two Bézier curves.
#* Press the button {{Button|[[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineDecreaseKnotMultiplicity|B-spline decrease knot multiplicity]]}}.
#* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineDecreaseKnotMultiplicity.svg|24px]] Decrease knot multiplicity}}.


'''Note:''' Decreasing the multiplicity from 1 to 0 will remove the knot since the result would be a curve with an "edge" at the knot position (''C''<sup>0</sup> continuity) and this is not supported. (To create curves with an "edges", you can create two splines and connect them.)


<div class="mw-translate-fuzzy">
{{Docnav
{{Docnav/ru
|[[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]
|[[Sketcher_BSplineIncreaseKnotMultiplicity/ru|Увеличение кратности узлов]]
|[[Sketcher_SwitchVirtualSpace|Switch Virtual Space]]
|[[Sketcher_SwitchVirtualSpace/ru|Переключение виртуального пространства]]
|[[Sketcher_Workbench|Sketcher]]
|[[Sketcher_Workbench/ru|Верстак "Sketcher"]]
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconL=Sketcher_BSplineIncreaseKnotMultiplicity.svg
|IconR=Sketcher_SwitchVirtualSpace.svg‎
|IconR=Sketcher_SwitchVirtualSpace.svg‎
|IconC=Workbench_Sketcher.svg
|IconC=Workbench_Sketcher.svg
}}
}}
</div>


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Latest revision as of 07:35, 22 April 2024

Уменьшение кратности узлов

Системное название
Sketcher_BSplineDecreaseKnotMultiplicity
Расположение в меню
Sketch → B-сплйан инструменты эскиза → Уменьшение кратности узлов
Верстаки
Sketcher
Быстрые клавиши
Нет
Представлено в версии
0.17
См. также
Показать/скрыть кратность узлов B-сплайна, Увеличение кратности узлов

Описание

The Sketcher BSplineDecreaseKnotMultiplicity tool decreases the multiplicity of a B-spline knot.

Применение

  1. Select a B-spline knot.
  2. There are several ways to invoke the tool:
    • Press the Decrease knot multiplicity button.
    • Select the Sketch → Sketcher B-spline tools → Decrease knot multiplicity option from the menu.

Example

See Sketcher_BSplineIncreaseKnotMultiplicity

Notes

If you decrease the multiplicity of a knot to zero, the knot vanishes. Mathematically it then appears zero times in the knot vector, meaning there is no longer a basis function. Understanding this requires some math, but it will also be clear if you look at the multiplicity. For example a knot with multiplicity 0 on a B-spline with degree 3 means that at the position of the knot two Bézier pieces are connected with C3 continuity. So the third derivative should be equal on both sides of the knot. However for a cubic Bézier curve this means that both sides must be part of the same curve. There is then effectively no longer a knot connecting two Bézier curves.