Geometric continuity

Geometric continuity is a term from the field of geometric modeling and describes the quality of the contact between two flat curves or surfaces at a common point without considering the randomly selected (parameterized or implicit or explicit) representation of the curves or surfaces.

Curves with contact (circles and a straight line) ${\displaystyle G^{1}}$

Curves with contact (left: curves , right: ellipse with vertex curvature circles ) ${\displaystyle G^{2}}$
${\displaystyle y=x^{3},y=0}$

${\displaystyle G^{2}}$-continuous transition surface (green) between two tensor product Bezier surfaces

• ${\displaystyle G^{1}}$-Continuity between

- two curves at a common point means that both curves have the same tangent (see 1st picture). ${\displaystyle P}$ ${\displaystyle P}$

- two surfaces at a common point means that both surfaces at the point have the same tangent plane . ${\displaystyle P}$ ${\displaystyle P}$

• ${\displaystyle G^{2}}$-Continuity between

- two curves at a common point means that both curves have the same tangent and the same (oriented) curvature (see 2nd picture). ${\displaystyle P}$ ${\displaystyle P}$

- two surfaces at a common point means that both surfaces in the same tangent plane and have the same normal curvatures . ${\displaystyle P}$ ${\displaystyle P}$

The continuity can generally be defined. - Continuity means that both curves / surfaces only have contact in the relevant point. In practice - and continuous curves / surfaces play the most important role. For example, when a straight road transitions into a circular arc road, the transition should of course be at least tangential ( ). If the road goes tangentially directly into the circle, the driver has to jerk the steering wheel from the straight-ahead position to the circular movement. To make this more harmonious, the transition between straight line and circle is taken over by a transition curve that has contact with both the straight line and the circle. This cannot be achieved with part of a conic section (second degree curves). You have to use at least one curve of the 3rd degree (parameterized or implicit) for this. An example of the necessity of -contacts between surfaces comes from the automotive industry. A body is usually composed of several parts. If this combination is only made with tangential contact ( ), mirror images of objects get a kink at these points, which looks unfavorable. This can be avoided with continuous transitions. ${\displaystyle G^{n}}$ ${\displaystyle G^{0}}$ ${\displaystyle G^{1}}$ ${\displaystyle G^{2}}$ ${\displaystyle G^{1}}$ ${\displaystyle G^{2}}$ ${\displaystyle G^{2}}$ ${\displaystyle G^{1}}$ ${\displaystyle G^{2}}$

Comment on the pictures: The pictures each show independent curves with a common point at which the curves touch. In practice, part of one curve is continued at the point of contact by the second curve. The same applies to surfaces.

Flat curves

In order to be able to examine flat curves for geometrical continuity (contact), knowledge of elementary differential geometry is necessary.

Representations of curves

A plane curve is usually explained as the image of a real interval with respect to a continuous function . This definition provides one ${\displaystyle [a,b]}$ ${\displaystyle \mathbf {c} }$

• Parametric representation . ${\displaystyle \mathbf {x} =\mathbf {c} (t)=(x(t),y(t)),\ t\in [a,b]}$

If the function is n-times continuously differentiable , then the curve is called - continuous or curve of the class . ${\displaystyle \mathbf {c} }$ ${\displaystyle C^{n}}$ ${\displaystyle C^{n}}$

• A curve is called regular if the tangent vector is at every point . ${\displaystyle C^{1}}$ ${\displaystyle {\dot {\mathbf {c} }}(t)\neq \mathbf {0} }$

Examples:

1. ${\displaystyle x=\mathbf {c} (t)=(2\cos t,2\sin t),0\leq t<2\pi }$is the circle with radius 2 and center at the zero point. Since the following applies, this representation is regular. ${\displaystyle {\dot {\mathbf {c} }}(t)=(-2\sin t,2\cos t)\neq (0,0)}$
2. ${\displaystyle x=\mathbf {c} (t)=(t,t^{2}),t\in [-1,1]}$is part of the normal parabola . Since the representation is regular. ${\displaystyle y=x^{2}}$ ${\displaystyle {\dot {\mathbf {c} }}(t)=(1,2t)\neq (0,0)}$
3. ${\displaystyle x=\mathbf {c} (t)=(t^{3},t^{6}),t\in [-1,1]}$is part of the normal parabola . There is for , the display is not regular. ${\displaystyle y=x^{2}}$ ${\displaystyle {\dot {\mathbf {c} }}(t)=(3t^{2},6t^{5})=(0,0)}$ ${\displaystyle t=0}$

In special cases, the curve can be shown as a graph of the function . Such a representation is called explicit and writes , where is: ${\displaystyle x(t)=t}$ ${\displaystyle y(t)}$ ${\displaystyle y=f(x)}$ ${\displaystyle f(x)=y(x)}$

• explicit representation . ${\displaystyle y=f(x)}$

If there is a function, this representation is always regular. ${\displaystyle f}$ ${\displaystyle C^{1}}$

Examples:

1. ${\displaystyle y={\sqrt {4-x^{2}}},-2\leq x\leq 2}$is the upper semicircle with radius 2 and center at the zero point. The whole circle cannot be shown explicitly.
2. ${\displaystyle y=x^{2},\ -1\leq x\leq 1}$ is part of the normal parabola.

A third, essential representation of a plane curve is the implicit representation. The curve is seen as part of the solution set of an equation . The justification for this is provided by the theorem of the implicit function . It says that under certain conditions an equation can be solved locally for y or x, i.e. H. there is an explicit representation locally. ${\displaystyle F(x,y)=0}$ ${\displaystyle F(x,y)=0}$

• implicit representation . ${\displaystyle F(x,y)=0}$

An implicit representation of a curve is regular if it is differentiable at every point of the curve and holds true. ${\displaystyle F(x,y)=0}$ ${\displaystyle F}$ ${\displaystyle (F_{x}(x,y),F_{y}(x,y))\neq (0,0)}$

Examples:

1. ${\displaystyle F(x,y)=x^{2}+y^{2}-4=0}$ implicitly describes the circle with radius 2 and center at the zero point.
2. ${\displaystyle F(x,y)={\sqrt {x^{2}+y^{2}}}-2=0}$ implicitly describes the circle with radius 2 and center at the zero point.
3. ${\displaystyle F(x,y)=x^{2}-y=0}$ implicitly describes the normal parabola.

A plane curve can therefore be described with the help of various representations.

For the composition of curves (geometric modeling) only the geometric properties tangent, curvature, ... play a role. So you need suitable formulas for this.

Tangent of a plane curve

The tangent in a curve point can be represented parameterized by ${\displaystyle \mathbf {p} _{0}}$

• ${\displaystyle \mathbf {x} =\mathbf {p} +u\mathbf {r} }$with a suitable direction vector . ${\displaystyle \mathbf {r} }$

A direction vector one in a point one

• parameterized regular curve is , ${\displaystyle \mathbf {x} =\mathbf {c} (t)=(x(t),y(t))}$ ${\displaystyle \mathbf {r} ={\dot {c}}(t)=({\dot {x}}(t),{\dot {y}}(t))}$
• explicit curve is , ${\displaystyle y=f(x)}$ ${\displaystyle \mathbf {r} =(1,f'(x))}$
• implicit curve is . ${\displaystyle F(x,y)=0}$ ${\displaystyle \mathbf {r} =(-F_{y}(x,y),F_{x}(x,y))}$

Curvature of a plane curve

The curvature of a plane curve indicates how quickly the unit tangent changes along the curve. For a straight line is the curvature , for a circle with radius r is the curvature . With a level curve, there are only two possible directions in which a curve can curve: left or right. The amount of curvature ${\displaystyle \kappa }$ ${\displaystyle \kappa =0}$ ${\displaystyle \kappa =1/r}$

• a parameterized curve is , ${\displaystyle (x(t),y(t))}$ ${\displaystyle \kappa (t)={\frac {{\dot {x}}(t){\ddot {y}}(t)-{\ddot {x}}(t){\dot {y}}(t)}{{\big (}{\dot {x}}(t)^{2}+{\dot {y}}(t)^{2}{\big )}^{3/2}}}}$

• an explicit curve is , ${\displaystyle y=f(x)}$ ${\displaystyle \kappa (x)={\frac {f''(x)}{\left(1+f'(x)^{2}\right)^{3/2}}}}$

• an implicit curve is ${\displaystyle F(x,y)=0}$ ${\displaystyle \kappa =-{\frac {F_{y}^{2}F_{xx}-2F_{x}F_{y}F_{xy}+F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}.}$

C ⁿ contact of plane curves

Definition: Two regular plane parameterized curves have -contact at a common point , if the derivatives of the functions up to order n agree at the point . ${\displaystyle \mathbf {x} =\mathbf {c} _{1}(t),\ \mathbf {x} =\mathbf {c} _{2}(t)}$ ${\displaystyle C^{n}}$ ${\displaystyle \mathbf {x} _{0}}$ ${\displaystyle \mathbf {c} _{1},\mathbf {c} _{2}}$ ${\displaystyle \mathbf {x} _{0}}$

The contact of two explicit or implicit curves is defined analogously . ${\displaystyle C^{n}}$

G ⁿ contact of plane curves

Definition: Two regular parameterized or explicit or implicit plane curves have -contact at a common point if there are regular parameterized (or explicit or implicit) representations locally for both curves that have -contact in . ${\displaystyle G^{n}}$ ${\displaystyle \mathbf {x} _{0}}$ ${\displaystyle C^{n}}$ ${\displaystyle \mathbf {x} _{0}}$

• In the case of -contact, it is sufficient to prove that the two curves have parallel tangent vectors in . Because then there is an arc length parameterization or with contact in for each curve . ${\displaystyle G^{1}}$ ${\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}}$ ${\displaystyle \mathbf {x} _{0}}$ ${\displaystyle \mathbf {x} =\mathbf {x} _{0}+s{\frac {\mathbf {v} _{1}}{|\mathbf {v} _{1}|}}+\cdots }$ ${\displaystyle \mathbf {x} =\mathbf {x} _{0}+s{\frac {\mathbf {v} _{2}}{|\mathbf {v} _{2}|}}+\cdots }$ ${\displaystyle C^{1}}$ ${\displaystyle \mathbf {x} _{0}}$

• In the case of -contact, it is sufficient to prove that the two curves have parallel tangent vectors and the same (oriented) curvatures in . Because then there is an arc length parameterization or with contact in for each curve . The vectors are normalized or perpendicular vectors. ${\displaystyle G^{2}}$ ${\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}}$ ${\displaystyle \kappa _{1},\kappa _{2}}$ ${\displaystyle \mathbf {x} _{0}}$ ${\displaystyle \mathbf {x} =\mathbf {x} _{0}+s{\frac {\mathbf {v} _{1}}{|\mathbf {v} _{1}|}}+{\frac {s^{2}}{2}}\kappa _{1}\mathbf {n} _{1}\cdots }$ ${\displaystyle \mathbf {x} =\mathbf {x} _{0}+s{\frac {\mathbf {v} _{2}}{|\mathbf {v} _{2}|}}+{\frac {s^{2}}{2}}\kappa _{2}\mathbf {n} _{2}\cdots }$ ${\displaystyle C^{2}}$ ${\displaystyle \mathbf {x} _{0}}$ ${\displaystyle \mathbf {n} _{1},\mathbf {n} _{2}}$ ${\displaystyle \mathbf {v} _{1}}$ ${\displaystyle \mathbf {v} _{2}}$

Conic section set: p fixed, variable ${\displaystyle \varepsilon }$

Example: The equations with fixed parameters and family parameters implicitly describe a family of conics with the common point (see figure). To determine whether the conic sections have contact at the origin, we form the partial derivatives: ${\displaystyle F(x,y)=(1-\varepsilon ^{2})x^{2}-2px+y^{2}=0\qquad ,\ p>0\,\varepsilon \geq 0}$ ${\displaystyle p}$ ${\displaystyle \varepsilon }$ ${\displaystyle (0,0)}$ ${\displaystyle C^{2}}$

${\displaystyle F_{x}=2(1-\varepsilon ^{2})x-2p,\ F_{y}=2y,\ F_{xx}=2(1-\varepsilon ^{2}),\ F_{xy}=0,\ F_{yy}=2}$.

The following applies at the zero point:

${\displaystyle F_{x}=-2p,\ F_{y}=0,\ F_{xx}=2(1-\varepsilon ^{2}),\ F_{xy}=0,\ F_{yy}=2}$.

Because of dependent, the curves have no -contact. ${\displaystyle F_{xx}=2(1-\varepsilon ^{2})}$ ${\displaystyle \varepsilon }$ ${\displaystyle C^{2}}$

If you calculate the curvature according to the above formula, you get . So is independent of the family parameter . This means that there are two conic sections of this family in the zero point contact. ${\displaystyle \kappa ={\frac {1}{p}}}$ ${\displaystyle \kappa }$ ${\displaystyle \varepsilon }$ ${\displaystyle G^{2}}$

Note: -Contact can also be defined for space curves in the same way. ${\displaystyle G^{n}}$

G ⁿ contact of surfaces

${\displaystyle G^{n}}$-Contact can be formally defined in the same way for areas. In the case of a surface, the curvature is replaced by the normal curvature.

Note: The assembly of curves / surfaces with -contact is relatively easy. Modeling curves with -contact is a little more difficult. It is quite difficult to create surfaces with contact. In the predominantly English-language literature, one can find the creation of continuous curves and surfaces under the title blending curves and surfaces . ${\displaystyle G^{1}}$ ${\displaystyle G^{2}}$ ${\displaystyle G^{2}}$ ${\displaystyle G^{n}}$

See also

• Non-Uniform Rational B-Spline

literature

• Gerald E. Farin: Curves and Surfaces in Computer Aided Geometric Design , Vieweg-Verlag 1994, ISBN 3-528-16542-1 , page 155, 272
• Gerald E. Farin, Josef Hoschek, Myung-Soo Kim (Eds.): Handbook of Computer Aided Geometric Design . Elsevier Science & Technology, 2002, ISBN 0-444-51104-0 , pp. 193 ( limited preview in Google Book search).
• Josef Hoschek: Fundamentals of geometric data processing , BG Teubner-Verlag 1989, ISBN 3-519-02962-6 , page 185, 277

Web links

• Geometry and Algorithms for COMPUTER AIDED DESIGN

Individual evidence

1. ↑ Geometry and Algorithms ... , op.cit., Page 55
2. ↑ Geometry and Algorithms ... , op.cit., Page 55
3. ↑ Geometry and Algorithms ... , op.cit., Page 119