Curvature is a measure of how much a curve deviates from being a straight line. It is defined as the reciprocal of the radius of curvature at any given point on the curve.
The curvature of a curve can be calculated using the formula K = |dT/ds|, where K is the curvature, T is the unit tangent vector, and ds is the arc length parameter. Alternatively, it can also be calculated as the second derivative of the curve's equation with respect to the arc length parameter.
Curvature is an important concept in mathematics and physics, and is used to describe the shape of curves and surfaces. It has practical applications in fields such as engineering, computer graphics, and robotics.
The tangential angle, also known as the angle of inclination, is the angle formed between a curve and a tangent line drawn at a specific point on the curve. It represents the slope or steepness of the curve at that point.
The tangential angle and curvature are related through the formula K = tan(θ) / r, where K is the curvature, θ is the tangential angle, and r is the radius of curvature. This means that the curvature is directly proportional to the tangential angle, and inversely proportional to the radius of curvature.