Autonomous Navigation for Flying Robots Lecture 2.2: 2D Geometry

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Presentation transcript:

Autonomous Navigation for Flying Robots Lecture 2.2: 2D Geometry Daniel Cremers Technische Universität München

Geometric Primitives in 2D 2D point Augmented vector Homogeneous coordinates Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots Homogeneous Vectors Homogeneous vectors that differ only by scale represent the same 2D point Convert back to inhomogeneous coordinates by dividing through last element Points with are called points at infinity or ideal points Daniel Cremers Autonomous Navigation for Flying Robots

Geometric Primitives in 2D 2D line 2D line equation Daniel Cremers Autonomous Navigation for Flying Robots

Geometric Primitives in 2D Normalized line equation where and is the distance of the line to the origin Daniel Cremers Autonomous Navigation for Flying Robots

Geometric Primitives in 2D Line joining two points Intersection point of two lines Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots 2D Transformations Translation where is the translation vector, is the identity matrix, and is the zero vector Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots 2D Transformations Rigid body motion or Euclidean transf. (rotation + translation) or where is orthogonal ( ) with Distances (and angles) are preserved Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots 2D Transformations Scaled rotation/similarity transform or Preserves angles between lines Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots 2D Transformations Affine transform Parallel lines remain parallel Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots 2D Transformations Homography or projective transf. Note that is homogeneous (only defined up to scale) Resulting coordinates are homogeneous Straight lines remain straight Daniel Cremers Autonomous Navigation for Flying Robots

Autonomous Navigation for Flying Robots Lessons Learned 2D points and 2D lines Homogeneous coordinates 2D transformations Next: Example Daniel Cremers Autonomous Navigation for Flying Robots