Thread: curve definition

can anyone tell me the definition of curve?

Of a curve or of curvature?

i for one think is kinda a dumb question not trying to insult you lol

Indeed, it is not a stupid question: can you give us a mathematically precise definition using words and no pictures?

well a curve can be explained by this:I being a intervel of real numbers
and the curve a continuous mapping
x is topological space...

...

Think of a curve as being the trajectory of a particle as it moves through space. That, more or less, means the same thing as your definition, in which the interval I should be thought of as an interval of time.

just tell me the definition of a curve/curvature

numb3rs already posted the technical definition of a curve. My response contained an informal definition.

As for curvature, well, I'm going to assume you're interested in the curvature of a curve in, say, the plane or 3-dimensional space. Let me give an intuitive description of curvature of a curve in the plane. Imagine a curve in the plane as the trajectory of a particle moving through the plane, and consider the tangent vector at each point of its path (i.e., the direction the particle is headed at each point). Then the tangent vector rotates as you move along the path. The curvature is a measure of how wildly the vector rotates--the more it rotates, the more curved the curve is. This fits the informal notion of "curviness". If you want a technical definition, check out Wikipedia. Be warned the articles get very technical, but if you have any questions, feel free to bring them back here for discussion.

(If riverrat is lurking: this is more up your alley, so please jump in and take the lead if you feel so inclined!)

the length of an arc is defined as s=r * angle between the two r.where r is the segment. but i think this definition is wrong as when r moves along the arc the curve continuously changes and so though angle might be any value it is contained within two segments of different length meaning that we cannot find the length of an arc with the above formula.

That's the formula for the length of a circular arc, i.e. a section of a circle. It's most certainly correct, as a circle has constant curvature (so it looks the same at any point).

how do we find length of a curve which is not a circle

Do you use a line/contour integral?

Originally Posted by parag1973

how do we find length of a curve which is not a circle

Well, you could use the same technique, more or less...

Take a look at the Fibonacci sequence on a graph...

What would be the use of finding a length of a curve that is not part of a circle?

The only way you would find a curve is if you used a little bit of quadratics, which is just like stretching a linear equation...

If you wanted to find a curve, you could just write it down in a mathematical equation using what you have seen...

ex. y=x^2
OR
ex. y=(x^2)/3+14x

...

You could graph and find these...

As for finding the length of a curve,
ALL CURVES ARE INFINITE UNLESS THEY ARE IN A CIRCLE...

What you would be looking for is a PART of a curve...

I know... I use ellipses too much...

bit4bit: Yeah, it's basically the line integral over the constant function 1.

parag1973: You need calculus to be able to define and evaluate the lengths of curves. Let's assume our curve is in the plane is parameterized by a variable t, i.e. our curve is given by two functions x = f(t), y = g(t), t in the interval [0,1]. Then the length of the curve is:

₀∫¹ sqrt(f'(t)²+g'(t)²) dt

thacheezinator: There are a lot more curves than just quadratics out there. And finding the length of a noncircular curves is extremely useful in, say, architecture and engineering.

Thanks