28.3 : Curve Equations

Curves are essential geometric elements characterized by tangent distance, chord length, middle ordinate, and total arc length. These measurements are crucial in understanding a curve's geometric and spatial properties and are defined by the relationship between its radius and its central angle.

The tangent distance (T) refers to the straight-line measurement from the intersection point of two tangents to either the start or end of the curve. This distance is influenced by the curve's radius (R) and the central angle (I), which together define the geometric layout of the curve. 

The length of the chord (L.C.) represents the shortest straight-line distance between the curve's start and endpoints. It is derived by considering the curve's endpoints as forming the base of an isosceles triangle, with the radius as equal sides. The relationship between the central angle and this triangular configuration determines the chord length, showing how it varies with changes in the angle or radius.

The middle ordinate (M) is the maximum perpendicular distance from the chord to the curve, typically at the midpoint of the chord. This measure reflects the curve's depth, influenced by its radius and angular extent. The relationship between the offset at the midpoint and the curve's radius provides the value of the middle ordinate, emphasizing its dependence on the curve's geometry.

The total length of the curve (L), or arc length, is directly proportional to the radius and the central angle. This relationship establishes how the curve's angular extent determines its arc's physical span. Together, these parameters offer a comprehensive framework for analyzing a curve's dimensions and spatial properties in geometric and practical applications.