The rope curve, also known as the catenary curve, is a mathematical curve that is formed when a rope or chain is hung between two fixed points and allowed to hang freely under its own weight. It is a curve that is very similar to a parabola, but with a key difference: while a parabola is a symmetrical curve that opens upwards or downwards, the rope curve is not symmetrical. The rope curve is described by a mathematical equation known as the catenary equation. The equation is derived based on the forces acting on the rope or chain, and it takes into account the weight of the rope and the tension forces at each point. The equation can be written in different forms, but the most common form is y = a cosh(x/a), where y is the vertical position of the curve, x is the horizontal position, and a is a constant that determines the shape of the curve. The rope curve has many practical applications.
You're probably not going to notice the extra weight of this rope on a short sport climb, but you'll feel it in your pack or on longer routes.
This is great for soft catches on lead, but can be dangerous or even scary for a seconder, particularly if there are obstacles on a climb that you don t want to hit, like a block or slab. In this Perspective article, we propose that a major philosophical change would benefit this field, a proposition that is based on evaluation of situations in which rodent models have provided useful guidance and in which they have led to disappointments.
The rope curve has many practical applications. One of the most well-known examples is in the design of suspension bridges. The curve of the bridge's cables is carefully calculated using the catenary equation to ensure that the cables can support the weight of the bridge and any vehicles or pedestrians crossing it.
Coiling Rope in a Box
What is the longest rope length L of radius r that can fit into a box? The rope is a smooth curve with a tubular neighborhood of radius r, such that the rope does not self-penetrate. For an open curve, each endpoint is surrounded by a ball of radius r. For a box of dimensions $11\frac$ and rope with $r=\frac$, perhaps $L=\frac+\frac<\pi> \approx 1.3$, achieved by a 'U':
I know packing circles in a square is a notoriously difficult problem, but perhaps it is easier to pack a rope in a cube, because the continuity of the curve constrains the options? (I struggle with this every fall, packing a gardening hose in a rectangular tub.) I am more interested in general strategies for how to best coil the rope, rather than specific values of L. It seems that if r is large w.r.t. the box dimensions (as in the above example), no "penny-packing" cross-sectional structure is possible, where one layer nestles in the crevices of the preceding layer.
This is a natural question and surely has been explored, but I didn't find much. Edit 1. It seems a curvature constraint is needed to retain naturalness: The curve should not turn so sharply that the disks of radius r orthogonal to the curve that determine the tubular neighborhood interpenetrate. Edit 2 (26Jun10). See also the MO question concerning decidability. Edit 3 (12Aug10). Here is an observation on the 2D version, where a $1 1 2r$ box may only accommodate one layer of rope. If $k=\frac$ is an even integer, then I can see two natural strategies for coiling the rope within the box:
$\qquad \qquad \qquad \qquad \qquad$ Red is rope core curve, blue marks the rope boundary.
Interestingly, if I have calculated correctly, the length of the red rope curve is identical for the two strategies: $$L = 2 (k-1)[r \pi/2] + 2(k-1)^2 r \;. $$ For $r=\frac$, $k=8$ as illustrated, $L=\frac<7\pi> + \frac \approx 7.5$. (As a check, for $r=\frac$, $k=2$, and $L$ evaluates to $\frac<\pi>+\frac$ as in the first example above.)
- packing-and-covering
- mg.metric-geometry
- dg.differential-geometry
- open-problems
The rope curve is also used in architecture and design to create aesthetically pleasing shapes, such as the graceful curves of rooflines or hanging structures. In addition to its practical applications, the rope curve has fascinated mathematicians and scientists for centuries. It was first studied by the mathematician James Bernoulli in the early 18th century, and since then, many mathematicians have contributed to the understanding and exploration of the curve. The rope curve has been studied in the context of calculus, physics, and engineering, and its properties have been examined in great detail. Overall, the rope curve is a fascinating mathematical curve that has important practical applications and has been the subject of much mathematical research. Its unique shape and properties make it a subject of interest for mathematicians, scientists, and engineers alike..
Reviews for "Rope Curve Art: Inspiring Creations and Artists"
1. Sarah - 2/5 stars: I was really excited to read "The Roep Curwe" as I had heard such great things about it, but unfortunately, it did not live up to my expectations. The characters felt one-dimensional and their actions seemed forced at times. The plot was also quite predictable, and I found myself losing interest halfway through. Overall, I was disappointed with this book and wouldn't recommend it to others.
2. Michael - 2/5 stars: "The Roep Curwe" was a book that I struggled to connect with. The writing style felt disjointed and the pacing was inconsistent. The story jumped around without proper transitions, making it difficult to follow. Many of the characters were also unlikable and lacked depth. While the concept had potential, the execution fell flat for me.
3. Emily - 1/5 stars: I really wanted to enjoy "The Roep Curwe", but it was a complete letdown. The narrative was confusing and the world-building was lackluster. The author introduced too many subplots without properly resolving them, leaving me with a sense of frustration. The dialogue was also stilted and unnatural, making it hard for me to become invested in the story. Overall, I wouldn't recommend this book to anyone.
4. Jacob - 2/5 stars: "The Roep Curwe" had an interesting premise, but it failed to captivate me. The pacing was slow, and there were many unnecessary tangents that didn't contribute to the overall plot. The characters were underdeveloped, and their motivations were unclear. I found it hard to care about their fate. While I appreciate the effort put into this book, I was ultimately disappointed with the execution.