Grids can be based on existing geometry such as curves or surfaces For example we might have an array of curves that we would like our grid to follow
PanelingTools provides many functions to turn base geometry of points, curves, surfaces, and polysurfaces into an ordered 2-dimensional grid The grid is then
10 avr 2010 · Primitive/spline components create curves based on user input For example, with 'Curve' component, we can define a curve out of 4 input points
Chapter 3 includes an in-depth review of parametric curves with special focus on NURBS curves and the concepts of continuity and curvature It also reviews
+ To draw interpolate curves, we should define point array first • In this case, the point array should contain three points defined in the previous step
In the second step we array them based on a certain We supply a curve as the initial input geometry, radius of a circle as fixed number (though it is
Rhinoceros geometry is based on the NURBS mathematical model, which focuses on producing mathematically precise curves and freeform surfaces (in contrary to
24 mai 2022 · Step 1: Create 3 curves for generating a lofted surface in Rhino Step 2: In GH, click on the Surface tab > Freeform and choose the
the way If you lofted four curves with the recording on and then edited the control points of one of these curves, the surface geometry would update
Vector representation ................................................................................ 1
Position vector .................................................................................... 2
Vectors vs. points ................................................................................ 2
Vector length ...................................................................................... 3
Unit vector .......................................................................................... 3
Vector operations ..................................................................................... 4
Vector scalar operation ......................................................................... 4Vector addition .................................................................................... 4
Vector subtraction ............................................................................... 5
Vector properties ................................................................................. 6
Vector dot product ............................................................................... 7
Vector dot product, lengths, and angles .................................................. 8 Dot product properties ......................................................................... 9 Vector cross product ............................................................................ 9 Cross product and angle between vectors ............................................. 10 Cross product properties ..................................................................... 11Vector equation of line ............................................................................ 11
Vector equation of a plane ....................................................................... 13
Tutorials ................................................................................................ 14
Face direction .................................................................................... 14
Exploded box .................................................................................... 18
Tangent spheres ................................................................................ 24Matrix operations .................................................................................... 28
Matrix multiplication ........................................................................... 28
Identity matrix .................................................................................. 29
Transformation operations ....................................................................... 30
Translation (move) transformation ....................................................... 30 Rotation transformation ...................................................................... 31 Scale transformation .......................................................................... 33 Shear transformation ......................................................................... 33 Mirror or reflection transformation ....................................................... 34 Planar Projection transformation .......................................................... 35Parametric curves ................................................................................... 37
Curve parameter ............................................................................... 37 Curve domain or interval .................................................................... 38 Curve evaluation ............................................................................... 39 Tangent vector to a curve ................................................................... 40 Cubic polynomial curves ..................................................................... 40 Evaluating cubic Bézier curves ............................................................. 41NURBS curves ........................................................................................ 42
Degree ............................................................................................. 42
Control points ................................................................................... 42
Weights of control points .................................................................... 44Knots ............................................................................................... 44
Knots are parameter values ................................................................ 44Evaluation rule .................................................................................. 46
Characteristics of NURBS curves .......................................................... 46 Evaluating NURBS curves ................................................................... 49Curve geometric continuity ...................................................................... 51
Curve curvature ..................................................................................... 51
Parametric surfaces ................................................................................ 52
Surface parameters ........................................................................... 52 Surface domain ................................................................................. 54 Surface evaluation ............................................................................. 55 Tangent plane of a surface .................................................................. 55Surface geometric continuity .................................................................... 56
Surface curvature ................................................................................... 57
Principal curvatures ........................................................................... 57 Gaussian curvature ............................................................................ 58 Mean curvature ................................................................................. 58NURBS surfaces ...................................................................................... 59
Characteristics of NURBS surfaces ....................................................... 60 Singularity in NURBS surfaces ............................................................. 62 Trimmed NURBS surfaces ................................................................... 62Polysurfaces ........................................................................................... 63
Tutorials ................................................................................................ 65
Continuity between curves .................................................................. 65 Surfaces with singularity..................................................................... 71References .............................................................................................. 74
Notes .................................................................................................... 74
Figure (7): Vector scalar operation In general, given vector a =
Figure (10): Vector subtraction. In general, if we have two vectors, a and b, then a - b is a vector that is calculated
as follows: a =a × b = i (a2 ȴ b3) + j (a3 ȴ b1) + k (a1 ȴ b2) - k (a2 ȴ b1) - i (a3 ȴ b2) - j (a1 ȴ b3)
a × b = i (a2 ȴ b3 - a3 ȴ b2) + j (a3 ȴ b1 - a1 ȴ b3) + k (a1 ȴ b2 - a2 ȴ b1) a × b =Figure (19): Translate all box corner points. Rotation transformation This section shows how to calculate rotation around the z-axis and the origin point
using trigonometry, and then to deduce the general matrix format for the rotation. Take a point on x,y plane P(x,y) and rotate it by angle(b).Figure (21): Scale geometry Shear transformation Shear in 3-D is measured along a pair of axes relative to a third axis. For example, a
shear along a z-axis will not change geometry along that axis, but will alter it along x and y. Here are few examples of shear matrices:Figure (28): Normalized curve domain to be 0 to 1. An increasing domain means that the minimum value of the domain points to the
start of the curve. Domains are typically increasing, but not always.Figure (31): Tangents to a curve. Cubic polynomial curves Hermiteii and Bézieriii curves are two examples of cubic polynomial curves that are
determined by four parameters. A Hermite curve is determined by two end points and two tangent vectors at these points, while a Bézier curve is defined by four points. While they differ mathematically, they share similar characteristics and limitations. Figure (32): Cubic polynomial curves. The Bézier curve (left) is defined by four points. The Hermite curve (right) is defined by two points and two tangents. In most cases, curves are made out of multiple segments. This requires making what is called a piecewise cubic curve. Here is an illustration of a piecewise Bézier curve that uses 7 storage points to create a two-segment cubic curve. Note that although the final curve is joined, it does not look smooth or continuous.Figure (33): Two Bezier spans share one point. Although Hermite curves use the same number of parameters as Bézier curves (four
parameters to define one curve), they offer the additional information of the tangent curve that can also be shared with the next piece to create a smoother looking curve with less total storage, as shown in the following.