COMPUTER AIDED ARCHITECTURAL DESIGN
Workshop 15 Notes, Week of November 14 , 2017

VECTOR ARRAYS AND MAPPED OBJECTS TO 3D SURFACES

PART 1. 2D translation.

Within this the front view place a 1' x 2' corner point to corner point rectangular surface at the origin and one above it along the Y-Axis. In addition, place points along the X-Axis at 0,0,0 and 0,2,0 and also one at the lower left hand croner of the upper rectangle surface.




Now open Grasshopper  and put into place corresponding parameter components. For the upper rectanglular suface at point create a Geo component and pt as follows.




Similarly for the lower regtangular surface create a Geo component and two pt components:




Next, setup a number slider  and rename it as "numCopies" for both rectangular sufaces, with a range of  1 to 20 and a current value of 4.




2. Epand out the Grasshopper canvas window, and for the upper rectangular surface,  go to the vector tab and select the Vector 2 pt component.  Similarly create one fo the lower rectangular surface.




For the upper rectanglar surface go the "Crv"  tab,  add a divide curve component,  and connect it  to the "Crv" and "numCopies" components. Furthermore,  add and connect the "Vec2pt" input port "A"  to the original point at the lower left corner of the rectangle and input port "B" to the output port "P" of the  "Divide" (divide curve) component.




For the lower rectangle, connect the point parameters to the input ports "A"  and "B" of the "Vec2Pt" component.




3. Expand the visible Grasshopper canvas window area again, and now for the upper rectangular surface, go the the "Trns" (transform) tab and add a "Move" component. However, for the lower rectangular surface add a "AriLinear" ( linear array) component.as follows:.




For the upper rectangular surface connect the "Geo" parameter to the input port "G" of the "Move" component and the output port "V" of the "Vec2Pt" component to the input port "T" of the "Move" component. Accordingly, the original rectangular surface copies and translates along the horizontal line as follows:





Similarly,  for the lower rectangular surface, connect the "Geo" component to the input port "G" of the "AriLinear" component , the output port "V" of the "Vec2Pt"  component to the input port "D" of the "AriLinear" components, and the output port of the "numCopies" numerical slider to the input port "N" of the "AriLinear" component.  Accordingly, the original rectangular surface copies acccoring the vector determined by the initial two points on the X-axis as follows:




The dynamics of tansforming the upper rectangular surface and of translating the lower rangular surface are similar but the means are distinct from one other. For example, for the upper rectangular surface, you can rotate the line and increase the numbers of copies to 10, but for the lower rectangular surface, you can move the seond of the two points to adust the vector direction  and increase the number of copies to 10.



The same setup can also impact a translation of the rectangular surfaces within  to the X-Y plane. For example, for the upper rectangular surface place a curve in the X-Y plane and use it to replace the one attached to the "Crv" parameter. For the lower rectangular surface surface, move the second point from its original location on the X axis to new location in the postivie X-Y plane area. Increase the number of copies for both to 20,  and the result could be similar to the image below:

PART 2: 3D Vector Mapping of Objects to Surfaces.

We can apply a similar strategy to mapping an objecton the ground  to the UV coordinate system of a doubly curved surface and also taking advantage of surface normals.

1.  Create a 20 x 20 rectangular surface from the location 0, 0, 0  to the location 20, 20, 0 in the XY plane. 

2. Turn on the control points and then select the ones on it's lower left and upper right corner in the X-Y plane. Wtih the Gumball tool move the control points above the ground plane to create a simple saddle shape.

3 . Next place a point at at the location -2, -2, 0 and a solid cylinder of radius 0.2 at the same location. Move the cynlinder downward so that it is bisected by the ground plane.

Save the Rhino file, open Grasshopper and initiate a new Grasshopper file.

4. Within the Grasshopper  "Params" (parameter tab), add a "Geo" and "Pt" component and connect them to the Cylinder and to the point in Rhino respectively.

Similarly, add a surface component and connect it to the simple saddle shape.

Add two integer sliders ranging in value from 2 to 100 and labelled "U" and "V" for the surface coordinate system.

5. In the next step we establish a set of vectors from the  point at the center of the cylinder on the ground  to U V points along the saddle surface. Begin by going to the "Surface"  tab, and in the area labelled "Uti" (utility) select a "Divide Surface" component (to the upper left of the "Util" lable and place it in the canvas window. 

Connect  the "U" and "V"  numerical sliders to the "U" and "V" input ports of the "SDivide" (divide surface)  component, and also connect the "Srf" component to the "S" input port of the "SDivide" component, and see the points generated along the saddle shape. The grid consists 11 spaces along each side of the surface as defined by 12 points (there's always one more point than the number of spaces).

Similar to Part 1 and the previous linear array example go to the  "Vec" (vector) tab and place a point to point "Vec2Pt" vector component in the canvas window.

Connect the original "Pt" component that was connected the point located at the center of the cylinder to the input port "A" on the "Vec2Pt" component, and connect the output port "P" of the "SDivide" component to the input port "B" on the "Vec2Pt" component.. Note that the double line connecting port "P" to port "B" indicates a list of points coming off the saddle shape rather than a single point.

Now go to the "Trns" (transform) tab, and from above the "Euclidean" area add a "Move" component (the orange arrow symbol between two white dots). Attach the "Geo" component to the input for "G" of the "Move" component, and attach the output port "V" from the "Vec2Pt" component to the input port "T" of the "Vec2Pt" component. When completed, the cylinders will map to the U V points on the saddle shape. They will also bisect the saddle shape similarly to how the original cylinder is bisected by the point on the ground and at its center.

Upon closer examination, however, the cylinders are perpendiclar to the ground plane and not to the surface locations where they are mapped. In order to address this we can rotate them from their current orientation to the orientations of the surface normals. First, staying witin the "Trns" tabl, select the down arrow in the "Euclidean" and then the "Rotate" icon that has one vector rotating towards another one with input ports G, C, F, and T. Next, returning to the "Vec" tab and the icons that can be viewed by selecting the down area labelled "Vector" on the right hand side,  add a "unitZ" vector to the canvas window as highlighed in green below.

To activate the rotation of the cylinders, connect the output port "G" from the "Move" component to the input port "G" of the "Rotate" component, connect the output port "P" from the "Subdivide" component" to the input port "C" (Centers of rotation) of the "Rotate" component,  connect the output port "N" (surface Normals) to the input port  "T" (the vector to rotate To) of the "Rotate" component, and connect the output port "V" of the "UnitZ" component to the input port "F" (the the vector to rotate From) of the the "Rotate Component".