Planar kinematic geometry

© doc. Ing. Ivana Linkeová, Ph.D.


Planar kinematic geometry studies geometrical properties of curves - trajectories of moving points and envelopes of moving curves. Physical properties are not taken into consideration.

Planar motion is fully determined if the following figures are given:

  1. Two trajectories of two moving points.

  2. Two envelopes of two moving curves.

  3. One trajectory of one moving point and one envelope of one moving curve.

  4. Fixed and moving centrodes of motion.

The first three ways of planar motion determination are included in this study material. Only straight lines or circles are considered to be the trajectories of moving points, envelopes of moving curves or moving curve itself.

For all tasks, the graphical task setting and animated representation of task setting is given. Then, the trajectory of moving point, envelope of moving circle and envelope of moving straight line is solved, tangent lines to the generated trajectories and point of contact between generated envelope and moving curve including. If the sufficient demonstrability is preserved, the construction of fixed centrode and moving centrode is animated, too. The construction of moving centrode is based on inverse motion to a given motion.

Individual figures are distinguished by the following colours:

   - given component and corresponding figures: moving point A, trajectory τA, normal line nτA ; envelope (a), normal line n(a)
   - given component and corresponding figures: moving point B, trajectory τB, normal line nτB ; envelope (b), normal line n(b)
   - moving point M, ttrajectory τM, normal line nM, tangent line tM
   - moving straigth line m, normal line n(m), moving circle k
   - envelope (m),  point of contact T(m), branch of envelope (k), point of contact T(k)
   - branch of envelope (k*), point of contact T(k*)
   - fixed centrode p, centre of instantaneous rotation S on fixed centrode
   - moving centrode h, centre of instantaneous rotation (S) on moving centrode
   - axiliary components and corresponding figures