Student Number : 8800092K - MSc research report - School of Education - Faculty of Science / This research report explored the strategies which grade 12 learners employ to solve geometric problems. The purpose of this research was to gain an understanding of how grade 12 learners begin to solve geometric problems involving cyclic quadrilateral and tangent theorems. A case study method was used as the main research method. The study employed the van Hiele level’s of geometric thought as a method for categorising learners levels of understanding. Data about the strategies which learners recruit to solve geometric problems were gathered using learner-based tasks, semi-structured interviews and document analysis. From the data gathered, the following patterns emerged: learners incorrect use of theorems to solve geometrical problems; learners base their responses on the visual appearance of the diagram; learners “force “ a solution when one is not available; learners’ views of proof. Each of these aspects is discussed. The report concludes that learners strategies to solving geometric problems are based largely on the manner in which educators approach the solving of geometrical problems.

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Geometric constructions using ruler and compass are being solved for thousands of years. Humans are capable of solving these problems without explicit knowledge of the analytical models of geometric primitives present in the scene. On the other hand, most methods for solving these problems on a computer require an analytical model. In this thesis, we introduce a method for solving geometrical constructions with access only to the image of the given geometric construction. The method utilizes Mask R-CNN, a convolutional neural network for detection and segmentation of objects in images and videos. Outputs of the Mask R-CNN are masks and bounding boxes with class labels of detected objects in the input image. In this work, we employ and adapt the Mask R- CNN architecture to solve geometric construction problems from image input. We create a process for computing geometric construction steps from masks obtained from Mask R- CNN and describe how to train the Mask R-CNN model to solve geometric construction problems. However, solving geometric problems this way is challenging, as we have to deal with object detection and construction ambiguity. There is possibly an infinite number of ways to solve a geometric construction problem. Furthermore, the method should be able to solve problems not seen during the...

We describe a program called SketchIT capable of producing multiple families of designs from a single sketch. The program is given a rough sketch (drawn using line segments for part faces and icons for springs and kinematic joints) and a description of the desired behavior. The sketch is "rough" in the sense that taken literally, it may not work. From this single, perhaps flawed sketch and the behavior description, the program produces an entire family of working designs. The program also produces design variants, each of which is itself a family of designs. SketchIT represents each family of designs with a "behavior ensuring parametric model" (BEP-Model), a parametric model augmented with a set of constraints that ensure the geometry provides the desired behavior. The construction of the BEP-Model from the sketch and behavior description is the primary task and source of difficulty in this undertaking. SketchIT begins by abstracting the sketch to produce a qualitative configuration space (qc-space) which it then uses as its primary representation of behavior. SketchIT modifies this initial qc-space until qualitative simulation verifies that it produces the desired behavior. SketchIT's task is then to find geometries that implement this qc-space. It does this using a library of qc-space fragments. Each fragment is a piece of parametric geometry with a set of constraints that ensure the geometry implements a specific kind of boundary (qcs-curve) in qc-space. SketchIT assembles the fragments to produce the BEP-Model. SketchIT produces design variants by mapping the qc-space to multiple implementations, and by transforming rotating parts to translating parts and vice versa.

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Robots must plan and execute tasks in the presence of uncertainty. Uncertainty arises from sensing errors, control errors, and uncertainty in the geometry of the environment. The last, which is called model error, has received little previous attention. We present a framework for computing motion strategies that are guaranteed to succeed in the presence of all three kinds of uncertainty. The motion strategies comprise sensor-based gross motions, compliant motions, and simple pushing motions.

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Sammanfattning Syftet med denna studie är att synliggöra hur elever formulerar geometriska resonemang med utgångspunkt i vardagliga och vetenskapliga begrepp. Studiens syfte motiveras genom de svårigheter eleverna upplever på grund av detta fenomen. Utifrån studiens syfte har forskningsfrågan formulerats på följande vis: Hur formulerar elever geometriska resonemang med utgångspunkt i vardagliga och vetenskapliga begrepp? 15 elever från två årskurs femmor deltog i studien. Metoden som vi har valt i denna kvalitativa studie är tematisk analys. Resultatet analyserades med hjälp av Duvals (1995) teori om visualisering och kognitiva modell. I resultatet framgår det att elever som formulerar geometriska resonemang med enbart vardagliga begrepp och inte vetenskapliga begrepp begränsar sina möjligheter till att utveckla sina geometriska resonemang. Dessa elever tenderar även att ha större svårigheter att använda vetenskapliga begrepp för att beskriva deras geometriska konstruktioner. / Abstract The purpose of this study is to explore how do students formulate geometric reasoning based on every day and scientific concepts. The purpose of the study is motivated by the difficulties students experience due to this phenomenon. Based on the purpose of the study, the research question has been formulated as follows: How do students formulate geometric reasoning based on every day and scientific concepts? 15 students from two fifth graders participated in the study. The method we have chosen in this qualitative study is thematic analysis. The results were analyzed using Duval's (1995) theory of visualization and cognitive model. The results show that students who formulate geometric reasoning using only every day concepts and not scientific concepts limit their opportunities to develop their geometric reasoning. These students also tend to have greater difficulty using scientific concepts to describe their geometric constructions.

Triangle meshes have become an increasingly popular shape representation. Given the ease of standardization it allows and the proliferation of devices (scanners, range images ) that capture and output shape information as meshes, this representation is now used in applications such as virtual reality, medical imaging, rapid prototyping, digital art and entertainment, simulation and analysis, product design and development. In product development manipulation of mesh models is required in applications such as visualization, analysis, simulation and rapid prototyping. The nature of manipulation of the mesh includes annotation, interactive viewing, slicing, re-meshing, mesh optimization, mesh segmentation, simplification and editing. Of these editing has received the least attention. Mesh model often requires editing either locally or globally based on the application. With the increased use of meshes it is desirable to have formal reasoning tools that enable manipulation of mesh models in product development. The mesh model may contain artifacts like self-intersection, overlapping triangles, inconsistent normal’s of triangles and gaps or holes with or without islands. It is necessary to repair the mesh before further processing the mesh model. An automatic algorithm is proposed to repair and fill arbitrary holes while maintaining curvature continuity across the boundaries of the hole. The algorithm uses slices across the hole to first identify curves that bridge the hole. These curves are then used to find the surface patch that would fill the hole. The proposed algorithm works for arbitrary holes in any mesh model irrespective of the type of underlying surface and is able to preserve features in the mesh model that are missing in the input information. Since editing during product development is mostly feature based, an automatic algorithm to recognize shape features by directly clustering the triangles constituting a feature in a mesh model is proposed. Shape features addressed in the thesis are volumetric features that are associated with either addition or removal of a finite volume. The algorithm involves two steps – isolating features in 2D slices followed by a 3D traversal to cluster all the triangles in the feature. Editing a mesh model mainly implies editing local volumetric features in that model. An automatic algorithm is proposed for parametric editing of volumetric features in the mesh model. The proposed algorithm eliminates the need of original CAD model while manipulating any volumetric feature in the mesh model based on feature parameters. An automatic algorithm to manipulate global shape parameters of the object using the mesh model is developed. Global shape parameters include thickness, drafts and axes of symmetry. As the mesh models do not explicitly carry this information global editing of mesh models (other than for visualization) has not been attempted thus far. This thesis proposes the use of mid-surface to identify and manipulate global shape parameters for a class of objects that are classified as thin walled objects. Mid-curves are first identified on slices of the part and then the mid-surface is obtained from these mid-curves. Results of implementation are presented and discussed along with the scope for future work.