# 포락선(envelope)과 curve stitching

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## 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

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

## 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

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

## 메타데이터

### Spacy 패턴 목록

• [{'LEMMA': 'envelope'}]