"포락선(envelope)과 curve stitching"의 두 판 사이의 차이

2021년 2월 21일 (일) 21:35 기준 최신판

개요

"one-parameter family 에 있는 모든 곡선에 적어도 한 점에서 접하는 성질을 갖는" 곡선
이를 주어진 곡선의 family에 대한 포락선이라 부른다.
이러한 그림을 그리는 기술은 curve stitching 또는 string art 라는 이름으로 불리기도 함

포락선(envelope )

곡선들이 매개변수 t 에 의해 \(F(x,y,t)=0\) 로 주어진다고 가정하자.
이 곡선들에 대한 포락선은 다음 연립방정식에서 t를 소거하여 얻을 수 있다.

\[\left\{ \begin{array}{c} F(x,y,t)=0 \\ \frac{\partial F}{\partial t}(x,y,t)=0 \end{array} \right.\]

증명

포락선이 \(\mathbf{r}(t)=(x(t),y(t))\) 로 매개화되었다고 하자. \(F(x(t),y(t),t)=0\)가 성립한다.

주어진 \(t=t_0\)에 대하여, 포락선의 점은 \(\mathbf{r}(t_0)=(x(t_0),y(t_0))\) 로 주어진다.

한편, 점 \((x(t_0),y(t_0))\)에서, family의 곡선 \(F(x,y,t_0)=0\)에 대하여 \(\mathbf{n}(t_0)=\langle F_{x}(x(t_0),y(t_0),t_0),F_{y}(x(t_0),y(t_0),t_0) \rangle\)는 수직인 벡터가 된다.

따라서 \(\mathbf{r}'(t_0)=\langle x'(t_0),y'(t_0)\rangle\) 에 대하여 \(\mathbf{n}(t_0)\cdot \mathbf{r}'(t_0)=F_{x}(x(t_0),y(t_0),t_0)x'(t_0)+F_{y}(x(t_0),y(t_0),t_0)y'(t_0)=0\)이 성립한다.

\(F(x(t),y(t),t)=0\) 의 양변을 t로 미분하면,

\(F_{x}(x(t_0),y(t_0),t_0)x'(t_0)+F_{y}(x(t_0),y(t_0),t_0)y'(t_0)+F_t(x(t_0),y(t_0),t_0)=0\) 이므로, \(F_t(x(t_0),y(t_0),t_0)=0\)가 성립한다.

임의의 \(t=t_0\)에 대하여 성립하므로, 포락선의 매개방정식 \(\mathbf{r}(t)=(x(t),y(t))\)은 다음 연립방정식을 만족시킨다 \[\left\{ \begin{array}{c} F(x(t),y(t),t)=0 \\\frac{\partial F}{\partial t}(x(t),y(t),t)=0 \end{array} \right.\] ■

예1

파라메터 t에 대하여 다음과 같은 직선들을 생각하자\[\frac{x}{t}+\frac{y}{10-t}=1\quad, t=1,\cdots, 9\]

그림을 보면, 이 직선들에 접하는 곡선이 나타나는 것을 관찰할 수 있다.
포락선을 구하기 위해 위에서 언급한 결과를 이용하자\[F(x,y,t)=t^2 + t(y-x-10) + 10x\]\[\frac{\partial F(x,y,t)}{\partial t}=2t+ y-x-10\]
따라서 envelope은 다음 두 방정식에서 t를 소거함으로써 얻을 수 있다.

\[\left\{ \begin{array}{c} t^2 + t(y-x-10) + 10x=0 \\ 2t+ y-x-10=0 \end{array} \right.\]

이로부터 \(x^2-2 x y-20 x+y^2-20 y+100=0\) 를 얻는다.
이는 이차곡선(원뿔곡선) 으로 판별식 \(\Delta=b^2-4ac=4-4=0\) 인, 포물선이 된다.

예2: 어떤 타원들의 envelope

파라메터 \(0<t<1\)에 대하여 다음과 같은 타원들이 주어진다고 하자\[\frac{x^2}{t^2}+\frac{y^2}{(1-t)^2}=1\]
\(F(x,y,t)=(t-1)^2 (t-x) (t+x)-t^2 y^2\)
\(F_{t}(x,y,t)=-2 \left(2 t^3-3 t^2-t x^2-t y^2+t+x^2\right)\)
\(\left\{ \begin{array}{c} F(x,y,t)=0 \\ \frac{\partial F}{\partial t}(x,y,t)=0 \end{array} \right.\) 으로부터 다음의 두 관계식을 얻을 수 있다

\[\left\{ \begin{array}{c} y^2=(1-t)^3 \\ x^2=t^3 \end{array} \right.\]

t를 소거하면 \(x^{2/3}+y^{2/3}=1\) 를 얻는다.
이는 애스트로이드 (astroid) 가 된다

심장형 곡선

심장형 곡선(cardioid)

메모

http://playingwithmathematica.com/2011/04/27/curve-stitching-with-mathematica/
http://britton.disted.camosun.bc.ca/string_art/jbstringart.htm
http://www.wikihow.com/Create-a-Line-Design
베지에 곡선
http://en.wikipedia.org/wiki/B%C3%A9zier_curve#Quadratic_curves
예
parabolic line construction
http://demonstrations.wolfram.com/CircleChordEnvelope/
envelope
http://en.wikipedia.org/wiki/Envelope_(mathematics)
http://jwilson.coe.uga.edu/Texts.Folder/Envel/envelopes.html
envelope equation
http://www.sjsu.edu/faculty/watkins/envelopetheo.htm
Envelopes and String Art (Gregory Quenell) http://faculty.plattsburgh.edu/gregory.quenell/pubpdf/stringart.pdf

매스매티카 파일 및 계산 리소스

https://docs.google.com/leaf?id=0B8XXo8Tve1cxZmVkZjZhYTItYjhlNi00ZDA4LWE4OTItMDQyMjU5Yjk5ZWMz&sort=name&layout=list&num=50

수학용어번역

envelope - 대한수학회 수학용어집
- envelope - 포락선

리뷰, 에세이, 강의노트

Loe, Brian J., and Nathaniel Beagley. “The Coffee Cup Caustic for Calculus Students.” The College Mathematics Journal 28, no. 4 (September 1, 1997): 277–84. doi:10.2307/2687149.

노트

말뭉치

The envelope of a family of curves g(x, y, c) = 0 is a curve P such that at each point of P, say (x,y), there is some member of the family that touches P tangentially.^[1]
At the point of tangency the envelope curve and the corresponding curve of the family have the same slope.^[1]
This is a instance of the condition that was found above for the envelope of a family of curves.^[1]
The envelope of this family of curves is a curve such that at each point it touches tangentially one of the curves of the family (Figure \(1\)).^[2]
Eliminating the parameter \(C\) from these equations, we can get the equation of the envelope in explicit or implicit form.^[2]
Besides the envelope curve, the solution of this system may comprise, for example, singular points of the curves of the family that do not belong to the envelope.^[2]
To find the equation of the envelope uniquely, the sufficient conditions are used.^[2]
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope.^[3]
Classically, a point on the envelope can be thought of as the intersection of two "infinitesimally adjacent" curves, meaning the limit of intersections of nearby curves.^[3]
This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.^[3]
But these conditions are not sufficient – a given family may fail to have an envelope.^[3]
Example: the envelope of a circle with constant radius the centre of which describes a parabola is a curve parallel to the parabola.^[4]
The envelope can also be seen as the singular solution of the differential equation of which the curves ( G t ) are solutions.^[4]
Special case: the envelope of a family of lines is a curve for which this family is the family of the tangents.^[4]
Envelopes of lines can be physically produced thanks to tables of wires.^[4]
For example, two parallel lines are the envelope of the family of circles of the same radius having centres on a straight line.^[5]
My precalculus class recently returned to graphs of sinusoidal functions with an eye toward understanding them dynamically via envelope curves: Functions that bound the extreme values of the curves.^[6]
Near the end is a really cool Desmos link showing an infinite progression of periodic envelopes to a single curve–totally worth the read all by itself.^[6]
When you graph and its two envelope curves, you can picture the sinusoid “bouncing” between its envelopes.^[6]
Those envelope functions would be just more busy work if it stopped there, though.^[6]
The envelope follows the intersection of adjacent curves.^[7]
The curvature theories for envelope curve of a straight line in planar motion and envelope ruled surface of a plane in spatial motion are systematically presented in differential geometry language.^[8]
Why cannot the long-run marginal cost curve be an envelope as well?^[9]
The curve long run average cost curve (LRAC) takes the scallop shape, which is why it is called an envelope curve.^[9]
As shown in the following figure, the slopes of the short-run average cost curves leads to the attainment of LRAC which is a scallop shaped which is why it is called the envelope curve.^[9]
In this paper, an envelope curve-based coverage theory (ECCT) is proposed for the rapid computation of accumulative and continuous coverage boundary during a given period.^[10]
First of all, the application of envelope curve theory to satellite coverage problem is introduced.^[10]
Under this application background, inner envelope curves and outer envelope curves are proposed for continuous and accumulative coverage.^[10]
Compactly it can be said that an envelope of a family of curves in the plane is a curve that is tangent to each member of the family at some point.^[11]
Origin obtains the upper, lower, or both envelopes of the source data by applying a local maximum method combined with a cubic spline interpolation.^[12]
Professor Takashi Iwasa at Tottori University in Japan proposed a more straightforward and experimental structure method for estimating an envelope curve of wrinkled-membrane surface distortions.^[13]
Professor Iwasa commenced his experimental work by developing a formula for calculating the envelope curves of the membranes whose surfaces have been wrinkled due to the compressive loadings.^[13]
The envelope of a set of curves is a curve C such that C is tangent to every member of the set.^[14]
The concept of envelope is easily understood by looking at its graph.^[14]
When a family of curves are drawn together, their envelope takes shape.^[14]
Cycloid, formed by the envelope of its tangents, and osculating circles.^[14]
If you lock all envelope curves globally, they cannot be edited with the mouse.^[15]
I couldn’t find an envelope curve for Australian record rainfall so made one as shown below.^[16]
The next step is to calculate the envelope curve – a straight line on a log-log plot of rainfall against duration that provides an upper bound of the record rainfall depths.^[16]
Record rainfalls and envelope curves are also available the world (WMO, 2009) and for New Zealand (Griffiths et al., 2014).^[16]
The envelope curve I proposed for Australia (the green line) looks much too steep as it crosses the world curve.^[16]
As you drag, the overall length of the envelope changes—with all following nodes being moved.^[17]
When you release the mouse button, the envelope display automatically zooms to show the entire envelope.^[17]
You can, however, move nodes beyond the position of the following node—even beyond the right side of the envelope display—effectively lengthening both the envelope segment and the overall envelope.^[17]
For example, the family of circles of the same radius with centres on a straight line has an envelope consisting of two parallel lines.^[18]
For example, the envelope of spheres with the same radius and centres on a straight line is a cylinder.^[18]
The line of contact of the envelope with one of the surfaces of the family is called a characteristic.^[18]
I’m trying to emulate the response of the analog envelope on my Intellijel Atlantis.^[19]
Of course, I could just use the envelope on the Atlantis, but if I can do the envelope inside the A4, then I can P-lock it, have different presets ready to go, etc.^[19]
I’m specifically thinking of a Gaussian envelope on a TH1, so not really sophisticated.^[20]
One adaptation of the S-curve is known as the envelope S-curve , which takes into consideration successive generations of technologies that provide the same benefits.^[21]
The term "envelope" refers to the curve that connects the tangents of the successive individual S-shaped curves.^[21]
Try connecting the tangents of these curves to form an "envelope" and base the forecast on the extrapolation of the envelope curve.^[21]
The dotted line represents the envelope for these two S-curves which can be used to forecast future generations of microprocessors.^[21]
Given similar basin characteristics, a peak lying close to the envelope curve might occur at other basins in the same region.^[22]
A method for determination of blood velocity envelopes from image data is reported that uses Doppler-data specific heuristic to achieve high accuracy and robustness.^[23]
Comparisons with manually defined independent standards demonstrated a very good correlation in determined peak velocity values (r equals 0.993) and flow envelope areas (r equals 0.996).^[23]
This paper tests the applicability of classic envelopes curves to the hydrological conditions of Ceará.^[24]
(1945) formulated another mathematical equation for the calculation of the envelope curves.^[24]
Several other studies have evaluated the envelope curves as an estimator of maximum floods.^[24]
(2011) used envelope curves to determine the maximum floods and their probabilities of exceedance in unmonitored basins in the state of Minas Gerais, applying the methodology of Castellarin et al.^[24]

소스

메타데이터

위키데이터

ID : Q1060372

Spacy 패턴 목록

[{'LEMMA': 'envelope'}]

[ref_6f47d956-1] 1.0 ^1.1 ^1.2 The Envelope Theorem and Its Proof

[ref_d1d2702a-2] 2.0 ^2.1 ^2.2 ^2.3 Envelope of a Family of Curves

[ref_047c8227-3] 3.0 ^3.1 ^3.2 ^3.3 Envelope (mathematics)

[ref_016241c0-4] 4.0 ^4.1 ^4.2 ^4.3 Envelope of a family of plane curves

[ref_922e65b4-5] Envelope | mathematics

[ref_403d7e79-6] 6.0 ^6.1 ^6.2 ^6.3 Envelope Curves

[ref_c395ea06-7] Getting the envelope of a family of curves.

[ref_6ebab08a-8] Curvature Theory of Envelope Curve in Two-Dimensional Motion and Envelope Surface in Three-Dimensional Motion

[ref_ed7833c8-9] 9.0 ^9.1 ^9.2 Why the Long-Run Average Cost Curve is called an Envelope Curve? Why cannot the long-run marginal cost curve be an envelope as well?

[ref_56b9f267-10] 10.0 ^10.1 ^10.2 An envelope curve-based theory for the satellite coverage problems

[ref_9c0b160a-11] Brilliant Math & Science Wiki

[ref_83292947-12] Envelope (Pro Only)

[ref_faa5091d-13] 13.0 ^13.1 Simplified estimation method for envelope curve of wrinkled membrane surface distortions

[ref_fb66901c-14] 14.0 ^14.1 ^14.2 ^14.3 Envelope

[ref_c1587d0f-15] Locking All Envelope Curves

[ref_c9863882-16] 16.0 ^16.1 ^16.2 ^16.3 Envelope curve for record Australian Rainfall

[ref_7aac6fe8-17] 17.0 ^17.1 ^17.2 Use Logic Pro Sculpture envelope nodes

[ref_28539a24-18] 18.0 ^18.1 ^18.2 Encyclopedia of Mathematics

[ref_c7f4dfe4-19] 19.0 ^19.1 Envelope curve options: which is closest to a particular analog ADSR?

[ref_9e986871-20] Obtaining an envelope curve

[ref_4ecf842e-21] 21.0 ^21.1 ^21.2 ^21.3 MarketingProfs Forecasting Tutorial

[ref_daf974fd-22] Envelope Curves for Extreme Flood Events

[ref_1a5e73a6-23] 23.0 ^23.1 Determination of the envelope function (maximum velocity curve) in Doppler ultrasound flow velocity diagrams

[ref_0e94913e-24] 24.0 ^24.1 ^24.2 ^24.3 Regional envelope curves for the state of Ceará: a tool for verification of hydrological dam safety

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"포락선(envelope)과 curve stitching"의 두 판 사이의 차이

2021년 2월 21일 (일) 21:35 기준 최신판

목차

개요