Constantangle

Constantangle refers to a geometrical property in which a specific angle between two lines, line segments or rays remains invariant across a set of points or along a curve. In planar geometry, the locus of all points that subtend a given constant angle at the endpoints of a fixed segment AB is an arc of a circle. This circle – sometimes called a constant‑angle circle – is determined by the angle α, with the endpoints of the arc lying on the extensions of AB. When α equals 0° or 180°, the locus degenerates to the line segment itself; for other values, it yields a circle on either side of AB. The construction can also be phrased in terms of the inscribed angle theorem: if ∠APB = α for all points P on the locus, then A and B lie on the same circle that passes through P.

In differential geometry the constant‑angle condition is extended to curves and surfaces. A curve has constant angle if its tangent vector makes a fixed angle with a given direction or with the radial vector from a fixed point. Classical examples include the logarithmic spiral, where the tangent makes a constant angle with the radial line, and helices, which form a constant angle with the axis of a right circular cylinder. A surface is a constant‑angle surface if its normal vector keeps a fixed angle with a constant direction in space. Cylindrical surfaces, right circular cylinders, and helicoids are notable examples. Such surfaces arise in the study of minimal surfaces, isothermic surfaces, and in the design of architectural forms.

Applications of constant‑angle concepts appear in optics, where a light ray maintaining a constant elevation relative to a surface leads to constructive interference patterns, and in mechanics, where rods or cables that preserve a constant inclination during motion simplify analyses. The constant‑angle property also appears in computer graphics for curvature‑based shading and in structural engineering for the design of twisted beams and DNA‑like helices.