Sine curved surfaces and their application

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Background and Inspiration

The inspiration for my final project in the subject CAD4 – Rhino & Grasshopper, was the “SMU MEADOWS MUSEUM WAVE SCULPTURE” by Santiago Calatrava Architects and Engineers. I was impressed by the shape and the beauty, that was created just by switching two sine curves.

1 – Analysis

For the first part of my assignment, I started to analyze sine curves and the surfaces, that can be created by them and how changing small parameters can change the look of the whole shape.

For the analyzing, I have worked on three different base shapes, to show a further overview of the possibilities of working with sine curves. These three parts are

1) Sine curve surfaces

2) Sine curve surfaces on a base curve

3) Sine curve surfaces on a base circle

1 – Analysis – Sine curve surfaces

I created a script, where you can easily change all the relevant parameters for the sine shaped surfaces just by sliders.

In this graphic, you can see the modifications of changing the specified parameters:

The script:

Using the formula “expression designer” tool, to create the formula for a basic sine function and adding all the parameters with sliders. To make the curve go along a base, we are using the range with a multiple of pi, to determine the length of the surface. The Points and the Sine curve are now getting the information for the base from the range, but the z parameter of the curve is depending on the parameters from the formula.

To create a whole surface out of this sine curve, we are repeating this step three times (we can also have more parallel sine curves, to make the surface more complex). With moving curve 2 and 3, we have three parallels, that we can use to make a loft out of it.

To really analyze the reactions of a change on the shape, I have made some examples.

Changing the amplitude:

Changing the wave length:

Changing of postponement in vertical direction:

1 – Analysis – Sine curve surfaces on a base curve

The exact same analysis can be done, when the prerequisite is not only a base line, but any abstract curve. For this application example, the surface can be spanned over every area.

The script:

The principle of the script is the same, as it was for the case with only a straight base. We still have the formula, which provides the z coordinates for the curve. But instead of just a Range of numbers for the base, we set any curve we want as the base of our surface and we deconstruct this curve, to use the x and y parameters for our surface.

1 – Analysis – Sine curve surfaces on a base circle

For the third case, I went through the same analysis on a shape, that is created by three circles. It is again the same principle, we can change any small parameter to make a huge impact on the shape of our surface.

The script:

For this, we are working with the same script as for the curved base, we just replace the curves through three circles, the radius can be different for any project.

2 – Application

To give all of this analysis part a more practical sense, I worked on a concept, to make each of these surfaces a pavilion.

The concept of the script works on every case, on every constellation of parameters and can easily be used.

The roof will always be one of the surfaces, that we analyzed in the first part, while the construction will be seen in the following script.

The script:

For the application of our surface as a pavilion, we are using the complete script from the analysing part. We can choose, what shape or variation we want to have as our roof.

On that basis, we are creating an area underneath the surface. For that we connect the length of the beginning of the script and the distance of the base lines to create a rectangle underneath. Inside this rectangle we use “Populate Geometry” Tool to spread a lot of points inside this area (these points are the base for our columns). The slider controls how many points. On the same time we are working on the lofted surface and we are giving it a depth with the extrude tool.

To get the columns for our construction, we have to now project the points from the ground onto the surface and connect these points with lines.

To determine the point, where the column should divide into branches, we are evaluating the length of the lines and with the slider, we can set a point on a certain height. On the extruded Roof surface we are now again creating a lot of random points (“Populate Geometry”). So now we can search for the 3 (or 4 or 5) points (inside the roof) that are the closest to each dividing point. From the dividing point we are then spanning a line the 3 (or 4 or 5) closest points, to get the branches.

Then we only need to add the pipes for the column and the branches and our construction is done.

As the Populate Geometry and the Projecting tools are depending on the shape of the roof, we can connect this script to each of the analysing cases from part one.

For the Application part, we just bake the script, add some materials and shadows in Rhino and we have a much more realistic understanding of what can be done with sinusoidal curved surfaces.

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