포락선(envelope)과 curve stitching

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

관련된 항목들

• 포물선
• 네프로이드

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.

메타데이터

위키데이터

• ID : Q214728

Spacy 패턴 목록

• [{'LOWER': 'bézier'}, {'LEMMA': 'curve'}]
• [{'LOWER': 'bezier'}, {'LEMMA': 'curve'}]

노트

말뭉치

1. 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.
2. At the point of tangency the envelope curve and the corresponding curve of the family have the same slope.
3. This is a instance of the condition that was found above for the envelope of a family of curves.
4. 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 \(1\)).
5. Eliminating the parameter \(C\) from these equations, we can get the equation of the envelope in explicit or implicit form.
6. 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.
7. To find the equation of the envelope uniquely, the sufficient conditions are used.
8. 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.
9. 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.
10. This idea can be generalized to an envelope of surfaces in space, and so on to higher dimensions.
11. But these conditions are not sufficient – a given family may fail to have an envelope.
12. Example: the envelope of a circle with constant radius the centre of which describes a parabola is a curve parallel to the parabola.
13. The envelope can also be seen as the singular solution of the differential equation of which the curves ( G t ) are solutions.
14. Special case: the envelope of a family of lines is a curve for which this family is the family of the tangents.
15. Envelopes of lines can be physically produced thanks to tables of wires.
16. For example, two parallel lines are the envelope of the family of circles of the same radius having centres on a straight line.
17. 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.
18. 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.
19. When you graph and its two envelope curves, you can picture the sinusoid “bouncing” between its envelopes.
20. Those envelope functions would be just more busy work if it stopped there, though.
21. The envelope follows the intersection of adjacent curves.
22. 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.
23. Why cannot the long-run marginal cost curve be an envelope as well?
24. The curve long run average cost curve (LRAC) takes the scallop shape, which is why it is called an envelope curve.
25. 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.
26. 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.
27. First of all, the application of envelope curve theory to satellite coverage problem is introduced.
28. Under this application background, inner envelope curves and outer envelope curves are proposed for continuous and accumulative coverage.
29. 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.
30. Origin obtains the upper, lower, or both envelopes of the source data by applying a local maximum method combined with a cubic spline interpolation.
31. 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.
32. 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.
33. The envelope of a set of curves is a curve C such that C is tangent to every member of the set.
34. The concept of envelope is easily understood by looking at its graph.
35. When a family of curves are drawn together, their envelope takes shape.
36. Cycloid, formed by the envelope of its tangents, and osculating circles.
37. If you lock all envelope curves globally, they cannot be edited with the mouse.
38. I couldn’t find an envelope curve for Australian record rainfall so made one as shown below.
39. 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.
40. Record rainfalls and envelope curves are also available the world (WMO, 2009) and for New Zealand (Griffiths et al., 2014).
41. The envelope curve I proposed for Australia (the green line) looks much too steep as it crosses the world curve.
42. As you drag, the overall length of the envelope changes—with all following nodes being moved.
43. When you release the mouse button, the envelope display automatically zooms to show the entire envelope.
44. 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.
45. For example, the family of circles of the same radius with centres on a straight line has an envelope consisting of two parallel lines.
46. For example, the envelope of spheres with the same radius and centres on a straight line is a cylinder.
47. The line of contact of the envelope with one of the surfaces of the family is called a characteristic.
48. I’m trying to emulate the response of the analog envelope on my Intellijel Atlantis.
49. 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.
50. I’m specifically thinking of a Gaussian envelope on a TH1, so not really sophisticated.
51. 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.
52. The term "envelope" refers to the curve that connects the tangents of the successive individual S-shaped curves.
53. Try connecting the tangents of these curves to form an "envelope" and base the forecast on the extrapolation of the envelope curve.
54. The dotted line represents the envelope for these two S-curves which can be used to forecast future generations of microprocessors.
55. Given similar basin characteristics, a peak lying close to the envelope curve might occur at other basins in the same region.
56. 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.
57. 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).
58. This paper tests the applicability of classic envelopes curves to the hydrological conditions of Ceará.
59. (1945) formulated another mathematical equation for the calculation of the envelope curves.
60. Several other studies have evaluated the envelope curves as an estimator of maximum floods.
61. (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.