For nonplanar case, the proposed method gives improved accuracy after employing constraint weighting.

Experiments are performed in both simulated and real environments. An error contribution criterion of sonar measurement is proposed to weight the constraints, which further reduces the influence of noisy input. Different from the existing method which assumes the features being on the same plane in order to meet the pre-condition of quasi-homography transformation, our method allows the observation points to be distributed spatially (not limited to a plane), thus extending the proposed method to more general application environments. Each measurement value is formulated as geometrical constraint and the recovered scale can be expressed in closed-form. Vocabulary Equilateral Triangle – 3 congruent sides and angles Isosceles Triangle – 2 congruent sides and angles Scalene Triangle – No congruent sides or angles Įquilateral Triangle Isosceles Triangle Scalene Triangle Įxample Find the angles.In this paper, the scale ambiguity in underwater monocular visual odometry is solved with 2-D sonar measurements obtained from three viewpoints. Vocabulary Acute Triangle – All angles are less than 90° Right Triangle – Has one 90° angle Obtuse Triangle – Has one obtuse angle In the figure Line A B Find each angle measure ĥ-3 Triangles Triangle Sum Theorem – The angle measures of a triangle in a plane add to 180° Because of alternate interior angles, the following is true: Symbols Parallel Perpendicular Congruent Įxample 1 In the figure Line X Y Find each angle measure Properties of Transversals to Parallel Lines If two parallel lines are intersected by a transversal: The acute angles formed are all congruent The obtuse angles are all congruent And any acute angle is supplementary to any obtuse angle If the transversal is perpendicular to the parallel lines, all of the angles formed are congruent 90° angles Transversals to parallel lines have interesting properties The color coded numbers are congruent Railroad Tracks Perpendicular Lines – Lines that intersect to form Right Angles Transversal – A line that intersects two or more lines at an angle other than a Right Angle More Vocabulary Right Angle – Measures exactly 90 ° Acute Angle – Measures less than 90 ° Obtuse Angle – Measures more than 90 ° Complementary Angle – Angles that measure 90 ° together Supplementary Angle – Angles that measure 180 ° together Įxample 2 Name the following: Right Angle Acute Angle Obtuse Angle Complementary Angle Supplementary Angle Įven MORE Vocabulary Congruent – Figures that have the same size AND shape Vertical Angles Angles A & C are VA Angles B & D are VA If Angle A is 60 ° what is the measure of angle B? ĥ-2 Parallel and Perpendicular Lines Vocabulary Parallel Lines – Two lines in a plane that never meet, ex. Chapter 5 – Plane Geometry 5-1 Points, Lines, Planes, and Angles 5-2 Parallel and Perpendicular Lines 5-3 Triangles 5-4 Polygons 5-5 Coordinate Geometry 5-6 Congruence 5-7 Transformations 5-8 Symmetry 5-9 Tessellationsĥ-1 Points, Lines, Planes & Angles Vocabulary Point – Names a location Line – Perfectly straight and extends in both directions forever Plane - Perfectly flat surface that extends forever in all directions Segment – Part of a line between two points Ray – Part of a line that starts at a point and extends forever in one direction Įxample 1 Name four points Name the line Name the plane Name four segments Name five rays

Plane geometry