Disk B-spline curve provides an alternative shape modeling tool to standard control point based modeling techniques by extending control points to control disks, resulting in an easy and straightforward way of modeling regions or shapes with adjustable thickness. Due to this feature, disk B-spline curves are frequently applied in artistic shape modeling applications, such as brushstroke representation, calligraphy, and animation, but they are also suitable for engineering purposes. However, the boundary curves of the described shapes tend to have unintentional self-intersections or cusps, negatively affecting the usability and artistic value of the resulting shape and texture. In this paper, we introduce an iterative algorithm to give a solution to this problematic issue with the help of an approximating circle skinning method, while preserving the advantageous properties of the disk B-spline curves. Our method also results in smoother texturing of the shape around the more sharply curved sections.

Abstract QR code is a widely used format to encode information through images that can be easily decoded using a smartphone. These devices play a significant role in most people's everyday lives, making the encoded information widely accessible. However, decoding the QR code becomes challenging when significant deformations occur in the label. An easy and quick solution to keep the deformation on a minimum level is to affix the label that contains the QR code onto a developable surface patch of a 3D model. The perspective distortion that can appear is efficiently dealt with during the decoding process. In recent years an alternative method has emerged. In the work of Kikuchi et al., the QR code is embedded onto B‐spline surfaces of CAD models to give more freedom in providing additional information. This method was further improved and extended by Peng et al. embed QR codes onto the surface of general meshes. This paper introduces a solution to embed QR codes onto the surface of general meshes without densely triangulating the selected area of the QR code. It uses the deferred shading technique to extract the surface normals and the depth values around the QR code's user‐given centre. We propose two methods for automatically finding the projection direction even when highly curved areas are selected based on the retrieved information while rendering the model. Besides, we introduce two methods needing a projection direction and a QR code centre to determine a size for automatically embedding the QR code. We propose patterns for decreasing the carving depth of the embedded QR codes, and we use the Horizon‐Based Ambient Occlusion to speed up the engraving process. We validate our method by comparing our results to the outcomes of Peng et al.

Abstract While considering a mirror and light rays coming either from a point source or from infinity, the reflected light rays may have an envelope, called a caustic curve. In this paper, we study developable surfaces as mirrors. These caustic surfaces, described in a closed form, are also developable surfaces of the same type as the original mirror surface. We provide efficient, algorithmic computation to find the caustic surface of each of the three types of developable surfaces (cone, cylinder, and tangent surface of a spatial curve). We also provide a potential application of the results in contemporary free-form architecture design.

Providing additional information for parts or items usually means to enclose it next to the object or affix it on the component when that is possible. However, another solution is available by gaining the benefits of an additive manufacturing technology, 3D printing. This technology makes it possible to embed the additional information onto the surface of the items, for example, in the forms of QR codes. In the work of Kikuchi et al. (2018), the QR code is embedded into CAD models that consist of B-spline surfaces by grooving its dark regions to shadow them. The method proposed by Peng et al. (2019) optimized the modules of the QR code and the depth to carve its dark modules into any general mesh. However, embedding the QR code with these methods, in some cases, especially in case of highly curved surfaces, the QR code is deformed during the process of projection onto the surface. This deformation can highly restrict the readability of the QR code. In this paper, we propose an improved method to embed QR codes onto free-form surfaces by using a low-end consumer-level 3D printer. Our aim is to provide a robust method to project the QR code onto surfaces even with high curvature. We discuss the problematic cases for the works mentioned above, and we present a process to find an optimal position and direction of projection for the QR code to avoid deformations on highly curved surfaces. To validate our method, we compare our results with the outcomes of Kikuchi et al. (2018) and Peng et al. (2019).

Presents shape control methods for cubic B-spline and NURBS curves by the modification of their knot values and by the simultaneous modification of weights and knots. Theoretical aspects of knot modification are also discussed, concerning the paths of points on a curve and the existence of an envelope for the family of curves resulting from a knot modification for curves of degree k.