# Can a curve with singular point be a regular curve?

• Cauchy1789
In summary, the conversation discusses the concept of regularity in parameterized curves and the proof that a curve is regular everywhere except at a specific point. The formula for the curve is given and it is concluded that the curve is indeed regular except at t = a, which is not even in the interval (a,b). The concept of regularity and its application is also briefly explained.
Cauchy1789

## Homework Statement

Given a parameterized curve $$\alpha:(a,b)\rightarrow \mathbb{R}^2$$, show that this curve is regular except at t = a.

## Homework Equations

I know that according to the defintion that a parameterized curve $$\alpha: I \rightarrow \mathbb{R}^3$$ is said to be regular if $$\alpha'(t) \neq 0$$ $$\forall t \in I.$$

## The Attempt at a Solution

I have read that any curve which has a point where the tangent vector is zero cannot be a regular curve, so how is it even possible to just forget about that singular point in such a proof?

Best regards
Cauchy

Also, a isn't in your interval, so you're fine anyway?

Office_Shredder said:

Also, a isn't in your interval, so you're fine anyway?

Hi

First of all its suppose be t = p and $$p \in I$$ and the curve is defined as

$$\alpha(t) = (x(t),y(t))$$ a parameter curve.

having a singular point on a regular curve isn't that a contradiction?

Sincerrely

Cauchy

show that this curve is regular except at t = a.

Where a is not in (a,b) so it doesn't even fail the regularity condition.

Anywho, if someone says "Show a curve is regular everywhere except point p" it's like if someone said "show f(x)=|x| is differentiable except at 0" By definition, a differentiable function is differentiable everywhere, but you understand what they mean anyway. Same principle applies here.

## 1. What is a singular point on a curve?

A singular point on a curve is a point where the curve fails to be smooth, meaning that the curve and its derivative have different values at that point.

## 2. Can a curve have more than one singular point?

Yes, a curve can have multiple singular points. These can be isolated points or they can form a continuous curve themselves.

## 3. What is a regular curve?

A regular curve is a curve that is smooth and has no singular points. This means that the curve and its derivative have the same value at all points on the curve.

## 4. Can a curve with a singular point still be considered a regular curve?

No, a curve with a singular point cannot be considered a regular curve. The presence of a singular point means that the curve is not smooth and therefore does not meet the criteria for a regular curve.

## 5. Are singular points always undesirable on a curve?

Not necessarily. In some cases, singular points can be intentional and serve a purpose in the behavior of the curve. However, in most cases, singular points are considered undesirable as they can cause issues with the continuity and differentiability of the curve.

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