3D Curves and NURBS PDF

AS2209 CURVES AND NURBS Curves are very precise. Meshes can only approximate a real object; it can look good but is not precise and exact. Designers and engineers often use curves because they are defined by math and are exactly the correct shape. M...

AS2209 CURVES AND NURBS Curves are very precise. Meshes can only approximate a real object; it can look good but is not precise and exact. Designers and engineers often use curves because they are defined by math and are exactly the correct shape. Models that are created with the use of curves consume less memory space than those created using meshes. Bezier curve NURBS curve Bezier and NURBS. Image retrieved from Blender for Dummies Curves are divided into two (2) main types: Bezier curves and NURBS curves. Major graphic programs such as Photoshop, Illustrator, and After Effects have some implementation of these types of curves and editing capabilities. BEZIER CURVES Bezier curves were popularized by French engineer Pierre Bézier who applied these curves to design and manufacturing applications, especially for automobiles. The Bezier curve was one of the early methods used by Computer-Aided Design (CAD) applications that used splines to generate smooth-flowing curves. In image editing programs, Bezier curves are known as “paths” and are smooth. Bezier curves are also used in animation to outline and visualize movements, like in easing. They are also used for fonts. Bezier Curve. Image retrieved from donqmedia.net Anatomy of a Bezier Curve Control Points – The control points define the shape of the curve. They are similar to vertices in mesh objects. On Bezier curves, the curve always intersects with a control point. Control Point Handles – Every control point has a pair of handles that you can use to control the rotation and curve influence of the control point. 02 Handout 1 *Property of STI  [email protected] Page 1 of 4  AS2209 Curve Segments – Between two (2) control points is a segment mathematically defined by the location and rotation of the control points. In its simplest form, a complete curve consists of two (2) control points and a segment between them. A control point can have at most one segment going into it and one segment going out. There is no such thing as a control point that connects three or more segments. Editing Bezier Curves The most defining aspect of Bezier curves is the control points’ handles. The handle is not the control point; it is the “arm” that extends from the control point. All control points have two handles. If you select the control point, you select the control point and both of its handles. If you click only one of the handle dots, then you’re only controlling that one handle on the control point. The lack of precision with the ability to control the curve is one of the problems frequently experienced with Bezier curves. As the entire curve representation averages all the vertices of the Bezier curve, modifying any of the point and handles transforms the curve as a whole. NURBS NURBS stands for Non-Uniform Rational B-Spline. The control points in NURBS curves do not have handles similar to Bezier curves. By default, NURBS control points don’t normally even touch the curve shape itself. Instead, the control points are weighted to influence the shape of the curve. Polygons are created using lines (edges) and points (vertices) that create the polygonal mesh together. To create a smoother appearance, a large number of polygons are needed. However, in case of a highly detailed object, more subdivisions will be needed to increase polygon count significantly. During these circumstances, NURBS are ideal and efficient. An advantage of NURBS is that they create complex 3D shapes and surfaces starting from simple 2D splines, and, more importantly, they can be modified interactively after their creation. The user can interact with the basic curve object to refine the model generated until the final desired 3D shape is obtained. NURBS Curve Properties. Image retrieved from Blender for Dummies. 02 Handout 1 *Property of STI  [email protected] Page 2 of 4  AS2209 Properties of a NURBS Curve Control Point Weight – Each control point has a weight. The weight, which is a value greater than zero, influences how much that control point influences the curve. Knots – Knots are vectors that describe how the control points influence the resulting curve. Order – Order implies that the lower the order, the more the curve directly follows the lines between control points. And the higher the order, the smoother and more fluid the curve is as it passes the control points. NURBS Surfaces NURBS Surface. Image retrieved from dbse.ovgu.de NURBS surfaces or “patches” are created when a NURBS curve is extruded or when two (2) or more join one another across the 3D space. These NURBS “patches” can be used to slowly build your 3D model. The difference between NURBS surface and a polygonal object is that the polygon surface has a limited number of vertices, edges, and faces, which means that each vertex is accounted for by its transform. However, on a NURBS surface, the number of vertices of triangles varies as they are generated in real-time. SWEEPING TECHNIQUES The basic idea behind sweeping involves taking a 2D shape and “sweeping” it to create a solid volume. Sweeping techniques such as extruding and lathing are time-saving modeling techniques widely used to create detailed 3D objects efficiently. A cylinder can be considered an extruded circle, and a box is an example of an extruded square or a rectangle. Lathe (Revolve) Lathing creates a 3D surface by revolving the basic shape on an axis, which means that all objects created from lathing is symmetrical on the axis of rotation. A 3D surface generated by lathing is known as a “surface revolution.” If a closed surface is needed, the surfaces typically need to be swept by the complete 360°. 02 Handout 1 *Property of STI  [email protected] Page 3 of 4  AS2209 Loft Lofting involves creating an intricate 3D object using two (2) two-dimensional objects like curves and splines. Typically, with lofting, two splines are needed: One of these two splines serves as the cross-section or the shape spline, and the other one serves as the path spline serving as the path along which the shape can be extruded. Subdivision Modeling Subdivision refers to dividing further or a subordinate division. The idea behind subdivision modeling is to create a simpler model and create more organic, smoothened versions through subdividing. Subdivsion Modeling. Image retrieved from 3D Modeling & Animation: A Primer References Chandramouli, M. (2022). 3D Modeling & Animation: A Primer. Boca Raton, FL: CRC Press. Garofalo, E. (2022, January 24). What are NURBS curves in 3D modeling? MUO. Retrieved from https://www.makeuseof.com/what-are-nurbs-curves/ J, D. (2021). Nurbs vs. Bezier: What's the difference? 3DBiology.com. Retrieved from https://www.3dbiology.com/nurbs-vs-bezier-whats-the- difference/ Van Gumster, J. (2020). Blender for Dummies (4th ed.). Hoboken, NJ: John Wiley & Sons, Inc. 02 Handout 1 *Property of STI  [email protected] Page 4 of 4