Curves in Space: Singularities and Developments

This lecture explores the concept of curves in space, focusing on singularities and developments. It covers topics such as the rotation of the osculating circle, the Frenet frame, and extrinsic and intrinsic singularities. The instructor discusses the relationship between the osculating circle and the Frenet frame, as well as the properties of maximums and minimums on curves. The lecture also delves into the development curve and the developing curve, illustrating how they are related. Various exercises and examples are provided to deepen the understanding of these geometric concepts.

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MATH-126: Geometry for architects II

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

Curvature

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point.

Osculating circle

In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans (Latin for "kissing circle") by Leibniz.

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in 3-dimensional linear algebra. Classic Euler angles usually take the inclination angle in such a way that zero degrees represent the vertical orientation. Alternative forms were later introduced by Peter Guthrie Tait and George H.

Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space. In lay terms, a frame of reference is a system of measuring rods used by an observer to measure the surrounding space by providing coordinates. A moving frame is then a frame of reference which moves with the observer along a trajectory (a curve).

Rotation

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis.

Gothic Surfaces: Curvature, Development, and Stereotomy MATH-126: Geometry for architects II

Delves into the geometric principles of Gothic architecture, focusing on surface curvature and stereotomy techniques.

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Explores the Frenet-Serret formulas for curvature and torsion in three-dimensional space.

Spatial Curves: Singularities and Mechanisms MATH-126: Geometry for architects II

Explores spatial curves, singularities, and mechanisms, analyzing their geometric properties and applications.

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Covers the mechanics of slender structures, focusing on exact kinematics and equilibrium equations of rods.

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Explores spatial curves, focusing on intersections and singularities in architectural contexts.