To provide the student with a solid framework for understanding and applying a number of geometric shapes and techniques as they are used in engineering and architectural design contexts as exemplified below.

For mechanical engineers: Geometric construction of screws, screw pumps, hydraulic pumps, gear wheels, and roller bearings (helices, helicoids, cycloids, hypocycloids, spherical forms).

For ship building engineers: Propeller geometries via deformations of standard profiles and the construction of ship hulls.

For architectural engineers: Classical geometric concepts and basic operations for shape design and form description in plane and space.

To apply 2x2 and 3x3 matrices and their properties to analyze simple geometric constructions in plane and space and thereby obtain and practice the essential understanding of coordinate transformation techniques.

To define and to calculate precise modifications of a given geometric object.

To apply computer experiments as an integrated part of the course for illustrations, learning, and calculations.

• Calculate on a vectorial basis the area and volume
• Apply matrix calculus to construct and and analyze deformations of basic objects and explain the induced change in area and volume
• Find parametrizations of simple geometric objects in plane and space
• Calculate and explain the notions of area and volume for parametrized objects
• Apply simple parametrizations or other representations to construct triangulations of surfaces and domains in space and compare the respective areas and volumes
• Apply basic kinematic concepts to analyse simple motions in the plane
• Calculate and explain the notions of arclength and curvature for curves in the plane
• Apply extrudition, offsetting, and projection to construct new geometric objects from old ones
• Apply twisting, tapering, shearing, bending, and inversion on simple domains in space
• Analyse these deformations concerning surface area, volume, isoperimetry, curvature, etc.
• Recognize and suggest applications of geoemtric methods in architectural engineering, mechanical constructions, etc.
• Apply the respective methods and concepts in an individually chosen project exercise - possibly from a wide window of offered projects - and present the findings in a report

Vectors in plane and space.

2x2 and 3x3 matrices.

Triangles and tetrahedra as basic objects.

Coordinate transformations.
Matrix deformations.

Topological combinations in plane and space.

Triangulations, parametrizations, splines, and other representations.

Simple kinematic analysis concerning controlled motions along curves in plane and space.

Frenet-Serret basis, curvature and torsion for curves.

Various types of extruditions.

Construction and deformation of surfaces and solids.

Gaussian and mean curvature of surfaces.

Project exercises from applications within e.g. architecture, mechanical constructions, and 3D printing.

H. Pottmann et al.: Architectural Geometry, Bentley Institute Press 2007.

Fabricius Bjerre [Geometri] i udvalg.

S. Markvorsen, [Notes].

The course provides a basic foundation for the understanding of the geometric operations which are applied in FEM modelling, machine element design, architectural engineering, and sculptural design.

Please contact the teacher for information on whether this course gives the student the opportunity to prepare a project that may participate in DTU´s Study Conference on sustainability, climate technology, and the environment (GRØN DYST). More information http://www.groendyst.dtu.dk/kursustilmelding.aspx