Implicit surface

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation : F(x,y,z)=0. An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for or or . The graph of a function is usually described by an equation z=f(x,y) and is called an explicit representation. The third essential description of a surface is the parametric one: (x(s,t),y(s,t), z(s,t)) , where the -, - and -coordinates of surface points are represented by three functions x(s,t)\, , y(s,t)\, , z(s,t) depending on common parameters s,t . Generally the change of representations is simple only when the explicit representation z=f(x,y) is given: z-f(x,y)=0 (implicit), (s,t,f(s,t)) (parametric). Examples: #The plane x+2y-3z+1=0. #The sphere x^2+y^2+z^2-4=0. #The torus (x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0. #A surface of genus 2: 2y(y^2-3x^2)(1-z^2)+(x^2+y^2)^2-(9z^2-1)(1-z^2)=0 (see diagram). #The surface of revolution x^2+y^2-(\ln(z+3.2))^2-0.02=0 (see diagram wineglass). For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example. The implicit function theorem describes conditions under which an equation F(x,y,z)=0 can be solved (at least implicitly) for , or . But in general the solution may not be made explicit.…

In mathematics, an implicit surface is a surface in Euclidean space defined by an equation F(x,y,z)=0. An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for or or . The graph of a function is usually described by an equation z=f(x,y) and is called an explicit representation. The third essential description of a surface is the parametric one: (x(s,t),y(s,t), z(s,t)) , where the -, - and -coordinates of surface points are represented by three functions x(s,t)\, , y(s,t)\, , z(s,t) depending on common parameters s,t . Generally the change of representations is simple only when the explicit representation z=f(x,y) is given: z-f(x,y)=0 (implicit), (s,t,f(s,t)) (parametric). Examples: The plane x+2y-3z+1=0. The sphere x^2+y^2+z^2-4=0. The torus (x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0. A surface of genus 2: 2y(y^2-3x^2)(1-z^2)+(x^2+y^2)^2-(9z^2-1)(1-z^2)=0 (see diagram). The surface of revolution x^2+y^2-(\ln(z+3.2))^2-0.02=0 (see diagram wineglass). For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example. The implicit function theorem describes conditions under which an equation F(x,y,z)=0 can be solved (at least implicitly) for , or . But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult. If F(x,y,z) is polynomial in , and , the surface is called algebraic. Example 5 is non-algebraic. Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces. F(x,y,z)=0 where function F meets the necessary conditions of differentiability. The partial derivatives of F are F_x,F_y,F_z,F_{xx},\ldots . F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0, and a normal vector is \mathbf n(x_0,y_0,z_0)=(F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0))^T. \kappa_n = \frac{\mathbf v^\top H_F\mathbf v}{\|\operatorname{grad} F\|} is the normal curvature of the surface at a regular point for the unit tangent direction \mathbf v . H_F is the Hessian matrix of F (matrix of the second derivatives). 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. The electrical potential of a point charge q_i at point \mathbf p_i=(x_i,y_i,z_i) generates at point \mathbf p=(x,y,z) the potential (omitting physical constants) F_i(x,y,z)=\frac{q_i}{\|\mathbf p -\mathbf p_i\|}. The equipotential surface for the potential value c is the implicit surface F_i(x,y,z)-c=0 which is a sphere with center at point \mathbf p_i . The potential of 4 point charges is represented by F(x,y,z)=\frac{q_1}{\|\mathbf p -\mathbf p_1\|}+ \frac{q_2}{\|\mathbf p -\mathbf p_2\|}+ \frac{q_3}{\|\mathbf p -\mathbf p_3\|}+\frac{q_4}{\|\mathbf p -\mathbf p_4\|}. For the picture the four charges equal 1 and are located at the points (\pm 1,\pm 1,0) . The displayed surface is the equipotential surface (implicit surface) F(x,y,z)-2.8=0 . F(x,y,z) = {} & \sqrt{(x-1)^2+y^2+z^2}\cdot \sqrt{(x+1)^2+y^2+z^2} \\ & \quad \cdot \sqrt{x^2+(y-1)^2+z^2}\cdot\sqrt{x^2+(y+1)^2+z^2} \end{align} the constant distance product surface F(x,y,z)-1.1=0 is displayed. For two implicit surfaces F_0(x,y,z)=0, F_1(x,y,z)=0 (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter \mu \in [0,1] : F_\mu (x,y,z)=\mu F_1(x,y,z)+(1-\mu)\,F_0(x,y,z)=0 In the diagram the design parameter is successively \mu=0, \, 0.33, \, 0.66, \, 1 . )]] image (central projection) of an approximation of three tori.]] \Pi -surfaces can be used to approximate any given smooth and bounded object in R^3 whose surface is defined by a single polynomial as a product of subsidiary polynomials. In other words, we can design any smooth object with a single algebraic surface. Let us denote the defining polynomials as f_i\in\mathbb{R}[x_1,\ldots,x_n](i=1,\ldots,k) . Then, the approximating object is defined by the polynomial F(x,y,z) = \prod_i f_i(x,y,z) - r where r\in\mathbb{R} stands for the blending parameter that controls the approximating error. Analogously to the smooth approximation with implicit curves, the equation F(x,y,z)=F_1(x,y,z)\cdot F_2(x,y,z)\cdot F_3(x,y,z) -r= 0 represents for suitable parameters c smooth approximations of three intersecting tori with equations \begin{align} F_1=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+y^2)=0, \\[3pt] F_2=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(x^2+z^2)=0, \\[3pt] F_3=(x^2+y^2+z^2+R^2-a^2)^2-4R^2(y^2+z^2)=0. \end{align} (In the diagram the parameters are R=1, \, a=0.2, \, r=0.01. ) ==Further reading== Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, Thorpe: Elementary Topics in Differential Geometry, Springer-Verlag, New York, 1979,

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