In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
F(x,y,z)=0.
The graph of a function is usually described by an equation
z=f(x,y)
(x(s,t),y(s,t),z(s,t))
x(s,t),y(s,t),z(s,t)
s,t
z=f(x,y)
z-f(x,y)=0
(s,t,f(s,t))
Examples:
x+2y-3z+1=0.
x2+y2+z2-4=0.
(x2+y2+z2+R2-a2)2-4R2(x2+y2)=0.
2y(y2-3x2)(1-z2)+(x2+y2)2-(9z2-1)(1-z2)=0
x2+y2-(ln(z+3.2))2-0.02=0
The implicit function theorem describes conditions under which an equation
F(x,y,z)=0
If
F(x,y,z)
Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
Throughout the following considerations the implicit surface is represented by an equation
F(x,y,z)=0
F
F
Fx,Fy,Fz,Fxx,\ldots
A surface point
(x0,y0,z0)
F
(x0,y0,z0)
(0,0,0)
(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z0))\ne(0,0,0)
(x0,y0,z0)
The equation of the tangent plane at a regular point
(x0,y0,z0)
Fx(x0,y0,z0)(x-x0)+Fy(x0,y0,z0)(y-y0)+Fz(x0,y0,z0)(z-z0)=0,
n(x0,y0,z0)=(Fx(x0,y0,z0),Fy(x0,y0,z0),Fz(x0,y0,z
| T. | |
| 0)) |
In order to keep the formula simple the arguments
(x0,y0,z0)
\kappan=
| v\topHFv | |
| \|\operatorname{grad |
F\|}
is the normal curvature of the surface at a regular point for the unit tangent direction
v
HF
F
The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of a parametric surface.
As in the case of implicit curves it is an easy task to generate implicit surfaces with desired shapes by applying algebraic operations (addition, multiplication) on simple primitives.
See main article: article and Equipotential.
The electrical potential of a point charge
qi
pi=(xi,yi,zi)
p=(x,y,z)
| F | ||||
|
.
c
Fi(x,y,z)-c=0
pi
The potential of
4
| F(x,y,z)= | q1 |
| \|p-p1\| |
+
| q2 | |
| \|p-p2\| |
+
| q3 | + | |
| \|p-p3\| |
| q4 | |
| \|p-p4\| |
.
For the picture the four charges equal 1 and are located at the points
(\pm1,\pm1,0)
F(x,y,z)-2.8=0
A Cassini oval can be defined as the point set for which the product of the distances to two given points is constant (in contrast, for an ellipse the sum is constant). In a similar way implicit surfaces can be defined by a constant distance product to several fixed points.
In the diagram metamorphoses the upper left surface is generated by this rule: With
\begin{align} F(x,y,z)={}&(\sqrt{(x-1)2+y2+z2} ⋅ \sqrt{(x+1)2+y2+z2}\\ & ⋅ \sqrt{x2+(y-1)2+z2} ⋅ \sqrt{x2+(y+1)2+z2}) \end{align}
the constant distance product surface
F(x,y,z)-1.1=0
A further simple method to generate new implicit surfaces is called metamorphosis of implicit surfaces:
For two implicit surfaces
F1(x,y,z)=0,F2(x,y,z)=0
\mu\in[0,1]
F(x,y,z)=\muF1(x,y,z)+(1-\mu)F2(x,y,z)=0
\mu=0,0.33,0.66,1
\Pi
R3
fi\inR[x1,\ldots,xn](i=1,\ldots,k)
F(x,y,z)=\prodifi(x,y,z)-r
r\inR
Analogously to the smooth approximation with implicit curves, the equation
F(x,y,z)=F1(x,y,z) ⋅ F2(x,y,z) ⋅ F3(x,y,z)-r=0
c
| 2+y | |
| \begin{align} F | |
| 1=(x |
2+z2+R2-a2)2-4R2(x2+y2)=0,
| 2+y | |
| \\[3pt] F | |
| 2=(x |
2+z2+R2-a2)2-4R2(x2+z2)=0,
| 2+y | |
| \\[3pt] F | |
| 3=(x |
2+z2+R2-a2)2-4R2(y2+z2)=0. \end{align}
(In the diagram the parameters are
R=1,a=0.2,r=0.01.
There are various algorithms for rendering implicit surfaces,[2] including the marching cubes algorithm.[3] Essentially there are two ideas for visualizing an implicit surface: One generates a net of polygons which is visualized (see surface triangulation) and the second relies on ray tracing which determines intersection points of rays with the surface.[4] The intersection points can be approximated by sphere tracing, using a signed distance function to find the distance to the surface.[5]