# Curves and surfaces in computer aided geometric design - Yamaguchi F.

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s = a (1.11)

P=t. (1.12)

The equation of a tangent line can be expressed in terms of a parameter è

as:

R(u) = P(t0) + ut(t0). (1.13)

Here the parameter è is the distance from the tangent point.

At a point where a curve is regular, the tangent line is unique. However, at a singular point there occur various anomalous cases. Examples of singular points are shown in Figs. 1.11 and 1.12. At a cusp, as shown in Fig. 1.11, P

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1. Basic Theory of Curves and Surfaces

Fig. 1.11. Example of a singular point (1)

Fig. 1.12. Example of a singular point (2)

can become discontinuous or a 0 vector. Figure 1.12 shows examples in which line segments are expressed parametrically. With the expression used in Fig. 1.12(a), the line segments are regular over the entire interval. In contrast, in Fig. 1.12(b), although the line being expressed is the same two singular points occur within the interval. In this case, as t increases from the starting point t = 0 to /=(5 —j/5)/10, the point on the line progresses steadily toward the end point. At t = (5 —j/5)/10 x = j> = 0, so this point is a singular point. From t = (5-j/5)/10 to t = (5 + j/5)/10, as t increases the point on the line progresses back toward the starting point. t = (5 + j/5)/10 is another singular point; there the direction reverses again and the point on the line progresses toward the end point t= 1. The possibilities for such parameterization, the assigning of a relation between points on a line and a parameter, are numerous. It is desirable to choose a parameterization such that the curve is normal at as many points as possible.

1.2 Curve Theory

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Normal plane Fig. 1.13. The normal plane

Another type of singular point is one at which the curve degenerates to a single point.

A plane which passes through point P(t0) on a curve and is perpendicular to the tangent line to the curve at that point is called normal plane (Fig. 1.13). The equation of a normal plane is:

Consider a point P(t0) on a curve and a plane which passes through it and two points very close to it, P{t0 + A^t) and P(t0 + A2t). The plane that is approached in the limit as Axt and A2t approach 0 independently is called the osculating plane at point P{t0). The osculating plane is given by the following equation.

The brackets indicate a triple scalar product; P(t0) x P(to)^0. Equation (1.15) is the condition for the three vectors R — P(t0), P(t0) and P(t0) to lie in the same plane.

A general plane passing through the point P{t0) can be written in terms of its unit normal vector a as a ¦ (R — P(toj) = 0. The length of the perpendicular distance from the point P(t0 + At) on the curve P(t) to this plane is:

(Fig. 1.14). In the special case when a-P(t0) = 0, a-P(t0) = 0, that is, when the plane is the osculating plane, this length is a 3rd-order infinitesimal with respect to A t. Consequently, the osculating plane is the plane that best fits the curve at the point />(f0)-

The line that lies in the osculating plane, passes through the point P(t0) and is perpendicular to the tangent vector P(t0) at that point is called the principal normal The line that passes through point P(t0) and is perpendicular to the osculating plane is called the binormal, and the plane determined by the

(R-P(t0))P(t0) = 0.

(1.14)

lR-P(t0),P(t0), /*((„)] = 0.

(1.15)

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1. Basic Theory of Curves and Surfaces

Fig. 1.15. Relation between the normal plane, osculating plane and rectifying plane. 1 normal plane; 2 rectifying plane; 3 binormal; 4 principal normal; 5 tangent line; 6 osculating plane

tangent and the binormal is called the rectifying plane. These relations are shown in Fig. 1.15.

1.2.2 Curvature and Torsion

Let us consider the meaning of the second derivative P"(s) = d2P/ds2. From the definition of a derivative, we have:

ð.û= ,im '"fa+^bÃÛ (U6)

As -> 0 As

As shown in Fig. 1.16(a) and (b), in the limit as As—>0 the numerator /’'(so + zls) — P'(s0) is perpendicular to the tangent vector at the point P{s0) and points toward the center of curvature of the curve. It can be seen from

1.2 Curve Theory

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P'{ 5q + ^s)

P'iso)

(b)

Fig. 1.16. Geometrical relation between the center of curvature, P'(so) and P'(s0 + zls)

Fig. 1.16(b) that its magnitude is A6. Consequently, the magnitude of P"(s0) is:

ëî -/s i

|P"(s0)| = lim — = lim JL— = - = k.

As-+0 AS As-+0 AS Q

Here q is the radius of curvature and ê is the curvature. P" can be expressed in terms of them as:

P" = — n = êï. (1.17)

â

Here n is the unit vector pointing toward the center of curvature. Since P"(s) is a vector that has a magnitude equal to the curvature at point s and points toward the center of curvature, it is sometimes called the curvature vector. Curvature is the rate of turning of the unit tangent vector t with respect to the length of the curve s, in other words, a quantity that indicates how rapidly or slowly the curve is turning.

Next, let us find the relationship between the curvature vector P"(s) and the derivative vectors P and P with respect to the parameter t. From Eq. (1.9):

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