Sketcher BSplineIncreaseKnotMultiplicity/ru: Difference between revisions

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{{GuiCommand/ru
{{GuiCommand/ru
|Name/ru=Увеличение кратности узлов
|Name/ru=Увеличение кратности узлов
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|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/ru|Показать/скрыть кратность узлов B-сплайна]], [[Sketcher_BSplineDecreaseKnotMultiplicity/ru|Уменьшение кратности узлов]]
|SeeAlso=[[Sketcher_BSplineKnotMultiplicity/ru|Показать/скрыть кратность узлов B-сплайна]], [[Sketcher_BSplineDecreaseKnotMultiplicity/ru|Уменьшение кратности узлов]]
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<span id="Description"></span>
<span id="Description"></span>
==Описание==
==Описание==


The [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|24px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Sketcher BSplineIncreaseKnotMultiplicity]] tool increases the multiplicity of a [[B-Splines|B-spline]] knot.
Increases the multiplicity of a B-spline knot. (See [[B-Splines|this page]] for more info about B-splines).


<span id="Usage"></span>
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 pieces are connected are called knots. A knot on a spline with degree ''d'' and 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/>
==Применение==
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]]}}:

# Select a B-spline knot.
# There are several ways to invoke the tool:
#* Press the {{Button|[[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]}} button.
#* Select the {{MenuCommand|Sketch → Sketcher B-spline tools → [[Image:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] Increase knot multiplicity}} option from the menu.

==Example==

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 pieces are connected are called knots. A knot with multiplicity ''m'' on a B-spline with degree ''d'' means the curve to the left and right of the knot has at least an equal ''n'' order derivative (called ''C<sup>n</sup>'' continuity) where ''n = d - m''.

In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The knots have multiplicity 1.

The multiplicity is indicated by the numbers in round brackets. See [[File:Sketcher_BSplineKnotMultiplicity.svg|16px]] [[Sketcher_BSplineKnotMultiplicity|Show/hide B-spline knot multiplicity]].


[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
[[File:Sketcher_KnotMultiplicity_multiplicity1.png|400px]]
{{Caption|B-spline where both knots have the multiplicity 1.}}
{{Caption|B-spline where both knots have 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:
A multiplicity of 3 will change this B-spline so that even the first order derivatives are not equal (''C<sup>0</sup>'' continuity). Here is the same B-spline where the multiplicity of the knot on the left was increased to 3:


[[File:Sketcher_KnotMultiplicity_multiplicity3.png|400px]]
[[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.}}
{{Caption|Same B-spline with knot multiplicity 3. A control point was moved to show that the knot has ''C<sup>0</sup>'' continuity.}}


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:
A consequence of a higher multiplicity is that for the price of loosing continuity you gain local control. Meaning changing one control point will only affect the B-spline locally.


This can be seen in this example, where the B-spline with knot multiplicity 1 from the first image above was taken, and the second control point from the right was moved up. As a result the complete shape of the B-spline has changed:
[[File:Sketcher_KnotMultiplicity_locality.png|400px]]
{{Caption|Effect of locality due to different multiplicity.}}


[[File:Sketcher_KnotMultiplicity_locality1.png|400px]]
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.


After increasing the multiplicity of the knots to 2, moving the second control point from the right results in significant changes on the right side of the shape only:
<span id="Usage"></span>

==Применение==
[[File:Sketcher_KnotMultiplicity_locality2.png|400px]]

==Notes==


* Knot multiplicity can also be increased with [[Sketcher_BSplineInsertKnot|Sketcher BSplineInsertKnot]].
# Select a B-spline knot, either:
#* Press the button {{Button|[[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] [[Sketcher_BSplineIncreaseKnotMultiplicity|Increase knot multiplicity]]}}.
#* Use the menu {{MenuCommand|Sketch → Sketcher B-spline tools → [[File:Sketcher_BSplineIncreaseKnotMultiplicity.svg|16px]] Increase knot multiplicity}}.





Latest revision as of 07:34, 22 April 2024

Увеличение кратности узлов

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

Описание

The Sketcher BSplineIncreaseKnotMultiplicity tool increases 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 Increase knot multiplicity button.
    • Select the Sketch → Sketcher B-spline tools → Increase knot multiplicity option from the menu.

Example

B-splines are basically a combination of Bézier curves (nicely explained in this and this video). The points where two Bézier pieces are connected are called knots. A knot with multiplicity m on a B-spline with degree d means the curve to the left and right of the knot has at least an equal n order derivative (called Cn continuity) where n = d - m.

In this cubic B-spline (degree 3) there are 3 segments, meaning 3 curves are connected at 2 knots. The knots have multiplicity 1.

The multiplicity is indicated by the numbers in round brackets. See Show/hide B-spline knot multiplicity.

B-spline where both knots have multiplicity 1.

A multiplicity of 3 will change this B-spline so that even the first order derivatives are not equal (C0 continuity). Here is the same B-spline where the multiplicity of the knot on the left was increased to 3:

Same B-spline with knot multiplicity 3. A control point was moved to show that the knot has C0 continuity.

A consequence of a higher multiplicity is that for the price of loosing continuity you gain local control. Meaning changing one control point will only affect the B-spline locally.

This can be seen in this example, where the B-spline with knot multiplicity 1 from the first image above was taken, and the second control point from the right was moved up. As a result the complete shape of the B-spline has changed:

After increasing the multiplicity of the knots to 2, moving the second control point from the right results in significant changes on the right side of the shape only:

Notes