What makes a curve regular?

What makes a curve regular?

1. A line or surface that bends in a smooth, continuous way without sharp angles. 2. The graph of a function on a coordinate plane.

What does a regular curve mean?

A differentiable curve is said to be regular if its derivative never vanishes. ( In words, a regular curve never slows to a stop or backtracks on itself.) Two differentiable curves and. are said to be equivalent if there is a bijective map. such that the inverse map.

How do I show a smooth path?

Smooth path definition

1. A path ϕ is a continuous mapping from a segment of the real axis into the complex numbers; i.e. ϕ:[a,b]→C.
2. A path ϕ is smooth if it is a differentiable path, and furthermore, the derivative map ϕ′:[a,b]→C is continuous.
3. OK, now here is my path: ϕ:[−2,2]→C by ϕ(t)=t2+it3.
4. ϕ′(t)=2t+i3t2.

How do you determine the smoothness of a curve?

Smoothness of curves and surfaces Consider the segments either side of a point on a curve: G0: The curves touch at the join point. G1: The curves also share a common tangent direction at the join point. G2: The curves also share a common center of curvature at the join point.

What is a smooth parameterization?

plane-curves parametric. A curve defined by x=f(t),y=g(t) is smooth if f′(x) and g′(x) are continuous and not simultaneously zero.

How do you smooth a vector?

How to Smooth Out Your Vector Artwork

1. Click on the “Select” tool and click on your vector artwork.
2. Go to “Object,” “Path” and choose “Simplify.”
3. Set the “Curve Precision.” Choose a number between 0 percent and 100 percent to tell the editor how closely you want the path to follow the original shape.
4. Click the “Smooth” tool.

Is a line a smooth curve?

If you are plotting measured data, the only valid connecting curve between points is a straight line (or a line which is fitted to a function that comes from a physical model of the data). A smoothed curve implies that the data goes places where it has not been measured.

Can line graph be curved?

When you create a line graph in Excel, the lines are angled and have hard edges by default. You can easily change this to a curved graph with nice, smooth lines for a more polished look.

Which curve can also be a straight line?

Transcendental Curve – This curve is different from the algebraic curve. This curve might have many intersecting points which will be straight. These might also have an infinite number of inflexion points.

What is a piecewise smooth curve?

A piecewise smooth curve is a curve C that can be broken into finitely many smooth pieces C1 C2., Cn where the terminal point of one piece is the initial point of the next (Figure 13.2.9). For example, a curve formed by two or more sides of a rectangle or a polygon is piecewise smooth.

How do you know if a piecewise curve is smooth?

Definition: A curve in the complex plane is said to be a Piecewise Smooth Curve if there exists finitely many points $a = a_0, a_1., a_n = b$ with. < a_n = b$ called a Partition of , for which: 1) is infinitely differentiable on each open subinterval . 2) The derivative of on each closed subinterval are continuous.

What is a smooth piecewise function?

Intuitively, the notion of a piecewise smooth function is meant to capture the idea of a function whose domain can be partitioned locally into finitely many “pieces” relative on which smoothness holds, and continuity holds across the joins of the pieces. Here smoothness refers to continuous differentiability.

What is smooth curve in complex analysis?

A smooth curve is a curve z : [a, b] → C with a non-vanishing, continuous derivative such that each point is traversed only once (z is one-to-one), with the possible exception of a curve such that the endpoints match (z(a) = z(b)).

What is a rectifiable curve?

A rectifiable curve is a curve having finite length (cf. Line (curve)). We can think of a curve as an equivalence class of continuous maps γ:[0,1]→X, where two maps γ and γ′ are equivalent if and only if there is an homeomorphism φ of [0,1] onto itself such that γ′=γ∘φ.