Introduction

This article will explain and demonstrate Implicit Modeling in the following two parts.

1. Explanation of Implicit Modeling, modeling by TPMS (Triply Periodic Minimal Surface)
2. Implicit modeling with geometry as an argument

The modeling is mainly done using the Rhinoceros-Grasshopper plug-in Jellyfish.

In Chapter 1, we will explain implicit modeling and perform various modeling with TPMS.

What is Implicit Modeling?

A function in solved form with respect to one of the variables, such as \(y = f(x)\), is called an explicit function. On the other hand, a function that is not written in solved form, such as \(F(x, y) = 0\), is called an implicit function. Implicit modeling is a method of modeling geometry in terms of implicit functions.

For example, when modeling a circle, the implicit function is defined as follows

\(F(x,y)=x^2+y^2-R^2\)

Modeling is performed by defining regions with equality or inequality based on this implicit function.

For example, in the above implicit function, assuming \(R=5\) and generating a shape in the region \(F(x,y)=0\), the following circle is generated in the \(xy\) plane. (For convenience of visualization, the circle is given a thickness)

If \(F(x,y)\leq 0\), the interior is a filled circle.

If \(F(x,y,z)=x^2+y^2+z^2-R^2< 0\), then it is a sphere.

One of the advantages of implicit modeling is that boolean operations (addition and subtraction of shapes) can also be represented by implicit functions, making it difficult for operations to fail.

For example, if there are two shapes \(G_1,G_2\) defined by the implicit function \(F_1<0,F_2<0\), the shape represented by the implicit function \(F=\min(F_1,F_2)<0\) is the sum of \(G_1\) and \(G_2\), while the shape represented by the implicit function \(F=\max(F_1,-F_2)<0\) is \(G_1\) minus \(G_2\).

Modeling with implicit functions

Corrugated plate

If we define the implicit function by the sum of the trigonometric functions as follows, we get a Corrugated plate.

\(F(x,y,z)=|\cos (k_xx+\phi_x)+\cos (k_yy+\phi_y) +k_zz|-t<0\)

If you continuously change the parameters \(k_x,k_y,k_z,\phi_x,\phi_y\), you can see how the shape changes.

TPMS

If we make it an implicit function that also varies trigonometrically in the `\(z\) direction, the structure is a kind of TPMS, Schwarz P with thickness

\(\text{Schwarz P}:F(x,y,z)=|\cos x+\cos y+\cos z +s|-t<0\)

Other common examples of implicit modeling are Gyroid and Schwarz D. The implicit notation and geometry of each with thickness are shown below.

\(\text{Gyroid}：F(x,y,z)=|\sin x\cos y+\sin y\cos z+\sin z\cos x +s|-t<0\)

\(\text{Schwarz D}：F(x,y,z)=|\sin x\sin y\sin z+\sin x\cos y\cos z+\cos x\sin y\cos z+\cos x\cos y\sin z +s|-t<0\)

The eye-catching image is a thicker version of Gyroid, assigned to the Nature Architects logo N.

Similarly, Schwarz P, D can be easily assigned.

Modeling with extended TPMS implicit functions

We will do some modeling by extending the implicit functions of TPMS.

Changing the Scale of Change

Introducing the parameter \(k_x,k_y,k_z\), which determines the scale of change in the \(x,y,z\) direction, to Gyroid's implicit function, and setting different values for \(k_x,k_y,k_z\) will change the period in each direction.

\(F(x,y,z)=|\sin k_xx\cos k_yy+\sin k_yy\cos k_zz+\sin k_zz\cos k_xx +s|-t<0\)

Continuously varying thickness

You can also change the position where the thickness is applied by changing the thickness offset parameter \(s\).

If the parameter \(s\) is a function that varies with spatial coordinates, the position of the thickness can be changed continuously.

\(F(x,y,z)=|\sin k_xx\cos k_yy+\sin k_yy\cos k_zz+\sin k_zz\cos k_xx +s(x,y,z)|-t<0\)

Similarly, if the parameter \(t\), which determines the thickness, is a function, the thickness can be varied.

Create Gyroid with cylindrical and polar coordinates

You can also assign Gyroid to cylindrical or spherical shapes by changing the coordinate system to cylindrical or three-dimensional polar coordinates and defining the implicit function of Gyroid.

\(F(r,\theta,z)=|\sin k_r r\cos n_{\theta}\theta+\sin n_{\theta}\theta\cos k_zz+\sin k_zz\cos k_rr +s|-t<0\)

\(F(r,\theta,\phi)=|\sin k_r r\cos n_{\theta}\theta+\sin n_{\theta}\theta\cos n_{\phi}\phi+\sin n_{\phi}\phi\cos k_rr +s|-t<0\)

Complement different TPMS

Intermediate shapes can also be created by adding together different TPMS implicit functions to create a new implicit function.

\(\text{Gyroid}：F_G(x,y,z)=\sin k_xx\cos k_yy+\sin k_yy\cos k_zz+\sin k_zz\cos k_xx
\\
\text{Schwarz D}：F_D(x,y,z)=\sin k'_xx\sin k'_yy\sin k'_zz+\sin k'_xx\cos k'_yy\cos k'_zz+\cos k'_xx\sin k'_yy\cos k'_zz+\cos k'_xx\cos k'_zy\sin k'_zz
\\
F(x,y,z)=|aF_G(x,y,z)+(1-a)F_D(x,y,z)+s|-t<0\)

If the mixing parameter \(a\) is a function that varies with spatial coordinates, it can be varied continuously from Gyroid to Schwarz D.

Conclusion

In this chapter, we have explained implicit function modeling and demonstrated modeling with TPMS.

In the next chapter, we will discuss modeling by implicit functions with shape as an argument.