Surfaces from Points
Introduction
This tutorial is a continuation of Grasshopper 3D Grid from Surface. It creates a series of surfaces between 2 input surfaces. This can be useful in generating a deformed three dimensional grid of points, for example, where every cell is defined and deformed in three dimensions. There are a few ways of doing this, and this is only one example.
Surface Divide
To start, in Rhino, you draw 2 curves to be lofted (in Grasshopper) into a surface. By changing the control points of these these two curves, the surface responds. We have explained this technique before see:Grasshopper 3D Grid from Surface. Set the two curves from Rhino and use to loft between them.
To define the second surface, you do the same. Space the two surfaces apart. With this definition we will build a series of surfaces in between these two pre-defined surfaces that equally divides the distances between the two. If you alter the outer surfaces, the inner surfaces will respond as well.
In order to define the distances between the surfaces, we need to extract the same number of points on both surfaces and connect them to one another. We use to do this. By default the U and V values are set to 10. You can change this with a Number Slider.
In general, it is good practice (also see:Grasshopper "Good Practice") to define important parameters with a parameter container. This allows for a good overview of your file and easy re-use of parameters. To do so, right-click the parameter U and select Extract Parameter. A new integer container appears. The same container can be found in . Connect a Number Slider to control the value of the parameter.
In this case, we extract both U and V parameters and give them a Number Slider. Use the same U and V values for both Surface Divisions. This is important to make sure there is an equal number of points on both surfaces.
Now you have a number of points on both surfaces. In the next step we will connect the points and create a 3D grid of Points.
3D Grid of Points
If we want to create a series of surfaces between these two original ones at equal distance from one another we can draw a line between the two surfaces and divide this line equally.
First, with the component, we draw a line between the points on the surfaces. Since you've divided the surfaces with the same number of U and V values, the data structures correspond and the lines should easily find the correct points.
Then, use to divide the lines. If your surfaces are not planar and extruded in a single dimension, the length of each curve will vary and therefore the distance between points as well. However, the number of points should be the same everywhere, to be controlled with a Number Slider. This determines how many surfaces you will place in between the two original surfaces.
Three-Dimensional Data Tree
The data collection of points on the surfaces is a two-dimensional data-structure, organized by the number of U and V. The lines draw between the two surfaces also follow the same structure, comparable to the x and y in the Cartesian coordinate system. When we divide these lines, we introduce another dimension in the data-structure, comparable to the z in the Cartesian coordinate system. Now we have a three-dimensional data-tree.
You can also see this when you double-click the . Here you see that the tree has 3 dimensions, indicated by a center and 2 surrounding rings. In contrast, a two-dimensional data-tree has a center and 1 surrounding ring.
Usually we try to avoid three-dimensional data trees because it is incredibly difficult to work with. To do this, we can flatten the points from the original tree. Right-click the A and B inputs on the Line component and select Flatten. Now we got rid of the first dimension of the data-tree.
For visual purposes, we simplify the tree, using The Simplify Tree option eliminates any overlapping shared branches . The result is a simplified data tree structure.
To understand the data structure we can visualize it with the component. If you select the first (0) branch you will see a set of points along a line from surface to surface (this was the third dimension of the data tree, or comparable to the z-direction of Cartesian coordinate system).
The surfaces we would like to create run in the opposite direction of the data tree as it is currently. To obtain this, we can use . Now, checking the first (0) branch again, you'll see that the data tree is grouped by original U and V values, per surface. This is exactly what we need to create a Surface from Points.
Surface from Points
Next we use the to create surfaces from the 3D grid of points. Here we need to again identify the U. Note the difference in description of the U: on the Surface from Points component the U specifies the number of points in the {u} direction whereas on the Divide Surface component the U specifies the number of segments in the {u} direction. For us it is important that the defined U (for the Surface from Points component) matches the number of segments in the U direction specified. This is always U +1.
In the example on the right, you see that the U-segments (green dots) is defined as 7 (0-6), resulting in 8 (0-7) U-points (red dots).
Therefore, we add an expression to the U input on the Surface from Points component. Right-click the U and select Expression. Enter x+1. The x recognizes any numeric input. If you enter u+1 it won't work! As you increase the original U value, there will always be added 1 to this for the Surface from Points component.
