Unit Normal always pointing toward concave side

In summary, the unit normal vector will always point toward the side of concavity on a curve due to the fact that its direction is the derivative of the unit tangent, which always changes towards the concave side. This can be mathematically represented by looking at the Taylor series expansion of a curve defined by a parametric function, where both the curvature and the direction of the unit normal are derived from the quadratic term, resulting in the normal always pointing towards the concave side. Despite attempts to prove this, there are no known proofs on this concept.
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Physically/conceptually, I understand why the unit normal vector will always point toward the side of concavity on a curve. It's because the unit normal's direction is the derivative of the unit tangent, and the unit tangent's change in direction is always toward the concave side.

But how is this represented mathematically? I tried to prove it, but I'm stuck. I've looked, but I couldn't find any proofs about this. Does anyone know?
 
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  • #2
Think about a curve defined by a parametric function, and the Taylor series expansion of the functions at any point along the curve.

The curvature and the direction of the unit normal both come from the quadratic term in the Taylor series, so the normal always points to the "same" (concave) side of the curve.
 

Related to Unit Normal always pointing toward concave side

1. Why does the unit normal always point toward the concave side?

The unit normal is a vector perpendicular to the surface of a curve or object at a given point. In the case of a concave surface, the curvature is bending inward, and the unit normal must point in the direction of this curvature to accurately represent the shape of the surface.

2. Does this rule apply to all concave surfaces?

Yes, this rule applies to all concave surfaces, regardless of their size or shape. The direction of the unit normal is determined by the curvature of the surface at a specific point, not the overall shape of the surface.

3. How does the unit normal affect calculations involving concave surfaces?

The unit normal is an important factor in many calculations involving concave surfaces, such as determining the direction of forces acting on the surface or finding the rate of change in a given direction. Without considering the direction of the unit normal, these calculations would be inaccurate.

4. Can the unit normal ever point away from the concave side?

No, the unit normal will always point toward the concave side of a surface. This is a fundamental property of curvature and cannot be changed.

5. Are there any exceptions to this rule?

In mathematical and geometric terms, there are no exceptions to this rule. However, in certain real-world scenarios, the direction of the unit normal may not be as straightforward. For example, in a tornado, the unit normal may not point directly toward the concave side of the funnel cloud due to the chaotic and constantly changing forces at play.

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