3D Vector Modeling
Faculty sponsor: Mehrdad Simkani
University of Michigan-Flint
In Linear Algebra, we typically think of Vectors as 1-dimensional objects, having both a direction and a magnitude. For vectors in Second Life to model mathematical vectors, there are so many design and modeling considerations, such as what kind of primitives should be used to model a vector and how many of them should be considered. Primitives in Second Life are simple 3D shapes used as the basic building boxes for all objects. There are 8 types of primitives: the Box, the Cylinder, the Prism, the Sphere, the Torus, the Tube, the Ring, and the Sculpted Prim. We consider two vector models derived from a sphere: Dimple E 0.500 and Spherical Vector Model Path Cut E 0.500. Both vector models have been inspired by the antipodal nature of circles and spheres, which is the reason why their diameters are twice their radii. Check out this website about antipodes (/wiki/Antipodal_point)
We begin building in Second Life by opening the Build Tools Window and select the sphere icon if it’s not already green—it is selected when it is green.
手机报in语Figure 1
Figure 2
After we select the sphere and move our cursor outside the Build Tools Window to the ground or the floor, it turns into a magic wand and then we are ready to create a sphere. We click the mouse on the spot where we want to create the sphere.
皇室的
赏赐Any nonzero vector can be written in terms of its magnitude and the unit vector that is pointing in the same direction. Therefore, that unit vector represents its direction. The vector-magnitude-direction equation is →
→‖‖
‖‖
We can manipulate a primitive‘s modeling parameters and enc ode vector’s magnitude and direction, which it is representing in terms of its modeling parameters. Specifically, the vector’s magnitude is half of one of the vector model’s 3 dimensions, and the unit vector is represented by the Zero Rotation Vector (ZRV) times the vector model’s rotation. The vector model’s dimension that changes with the vector‘s magnitude that is being modeled is called the Active Dimension.
The Antipodal Nature of our Vector Models is Hidden
Our vector models are inspired by the antipodal nature of circles and spheres, like the Dimple E 0.50
0, the Spherical Vector Model Path Cut E 0.500, and also the Slice B 0.500 Vector Model. Even though, the Slice B 0.500 Vector Model isn’t derived from a sphere, it has an antipodal line
包仁segment in disguise like the Dimple E 0.500, the Spherical Vector Model Path Cut E 0.500. So their antipodal nature is hidden. If you have a vector model with its tail at the center of a sphere, the vector head would be a point on the surface of the sphere. Let the radius of the sphere be r and be equal to the magnitude of the vector:
‖‖√
The diameter is twice the magnitude of the vector, which is the same as the active dimension. Let
‖‖√
So the vector model’s active dimension changes as the vector’s magnitude changes. That’s why we call it “Active Dimension.”
One may ask why we put so much emphasis on the hidden antipodal nature of the vector model. The vector’s tail is the point where the vector rotates and scales from. Since primitives in Second Life like our vector models to rotate and scale from their geometric centers, the vector model’s tail ne
eds to be in the primitive’s geomet ric center. We think of a primitive being surrounded by a transparent rectangular prism that has 3 dimensions: x, y, and z. The corresponding dimensions are aligned to the corresponding regional axes x, y, and z, when the primitive has a zero inherent or implied rotation. A zero inherent rotation is a rotation when a primitive is rotated by zero degree around every regional axis: x, y, and z. Second Life is made of regions, and each region is a 256 meter by 256 meter square. More information about regions, can be found at: lslwiki/lslwiki/wakka.php?wakka=simulator
A region’s x-axis goes from west to east and a region’s y-axis goes from south to north. The most southwestern point in each region is (0, 0). Of course, the region’s z-axis goes from down to up. A region’s x-axis is color-coded red, a region’s y-axis is color-coded green, and a region’s z-axis is color-coded blue. The increasing end of the x-axis points to the east (to the right), the increasing end of the y-axis points to the north (into the screen), and the increasing end of the z-axis points up. The picture directly below is a sphere in Second Life being edited with its position handles, which are used to change its position. Each position handler is always parallel to its corresponding axis—it is color-coded as such, and each position handle points towards the increasing (positive) direction along its corresponding axis. In Figure 1-1, the s phere’s rotation is set to zero, which means that the yz68
angle around all 3 axes is zero, as we see in Figure 1-1. The sphere is rotated 90 degrees around the y-axis, as seen in Figure 1-2. The editing window has the 3 rotation fields, and the y-axis rotation is set to 90 degrees. So, the x dimension is aligned with the z-axis, and the z dimension is aligned with the x-axis. In Figure 1-3, the sphere has a zero rotation, so all of its dimensions are aligned with their corresponding axes. To make the Path Cut E 0.500 Vector Model, first we resize by changing its x and y dimensions from 0.5 meter to 0.1 meter. We change its z dimension from 0.5 meter to 2.0 meters. The editing window has 3 size fields, one for each dimension. The editing window has 2 Path Cut fields: The Path Cut
脏弹
B (Begin) field and the Path Cut E (End) field. We set the Path Cut E to 0.500. This renders the bottom half of its z dimension invisible and its z dimension is the vector m odel’s active dimension, so half of its z dimension represents the vector’s magnitude that the vector model represents.
豫公网安备41160202000603
站长QQ:729038198
关于我们
投诉建议