ELEMENTARY CURVES Curve (curve arch or segment) is any non-empty subset of space, which is a continuous image of an interval in real numbers. Analytic representation of curve is a vector function r(u) = (x(u), y(u), z(u)),
u∈ I defined, continuous and at least once differentiable on I.
Curve is a hodograph of this vector function r(u). Number a ∈ I is called parametric - curvilinear coordinate of point P(a), representing its position on curve. Functions x(u), y(u), z(u) are coordinate functions. Viviani curve is the trajectory of a point in the movement composed from two revolutions - about coordinate axis z
and simultaneously about coordinate axis x. Analytic representation of curve – vector function r(u) = (x(u), y(u), z(u)) for u ∈ I is equivalent to parametric equations of curve x = x(u), y = y(u), z = z(u) , u ∈ I Derivative r´(u) = (x´(u), y´(u), z´(u)) of vector function r(u) is vector function representing for a ∈ I direction vector of tangent to curve at the point P(a) r´(a) = (x´(a), y´(a), z´(a)). Orientation of this vector is the same as orientation of curve arc at the point P(a).
Point on curve, at which the orientation of the tangent direction vector is changing, is called cuspidal point. r´´(a) = (x´´(a), y´´(a), z´´(a)) = 0. Properties of curve represented at its regular points by means of derivatives of the curve vector function, and by length of vectors r´(a), r´´(a), or r´´´(a), are called intrinsic (geometric) properties. Unit tangent vector t(a) at the curve regular point P(a) is represented by formula Unit binormal vector b(a) is determined as vector product of the first and the second derivative of the curve vector equation r(u) at the curve regular point P(a) Line determined by point P(a) and direction vector b(a) is binormal. Unit (principal) normal vector n(a) at the curve regular point P(a) is a vector product of unit binormal and unit tangent vector n(a) = b(a) x t(a) Line determined by point P(a) and direction vector n(a) is principal normal. At any curve regular point P(a) there exist three perpendicular unit vectors t(a) ⊥ b(a) ⊥ n(a) ⊥ t(a) determining three perpendicular lines, t ⊥ b ⊥ n ⊥ t,
with common point P(a). Frenet-Serret moving trihedron at the curve regular point P(a) is a geometric figure
uniquely determined as intersection of three half-spaces with boundary planes ω, ρ, ν and direction vectors b, n, t (in given order). Edges are half-lines with common start point at the trighedron vertex P(a).
At the curve point of inflection holds 1k = 0. If 1k = 0 at all curve segment regular points, it is a line segment. Radius of the first curvature at the point P(a) is number 1ρ = 1/1k,
for 1k Osculting circle of curve k at the regular point P(a) is located in the osculating plane ω . Deviation of curve from the osculating plane at the non-inflection point P(a) is expressed as the second curvature of curve – torsion 2k. At the curve point of inflection is 2k = 0.
If 2k = 0 at all points on curve, curve is entirely located in one osculating plane, it is a planar curve. Rectification of a curve segment is development of curve at the neighbourhood of its regular point P(a) to the rectification plane at this point. Construction used to find a curve segment length is also called rectification. Curve segment length has to be calculated by formula Degree of a curve is natural number equal to maximal number of possible curve intersections with a line. 2. Cycloidal curves 3. Helices 4. Interpolation curves D. Velichová, 3DGeometry |