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Essential Constructions

Before diving into linkage synthesis, it's helpful to understand some fundamental geometric constructions using compass and straightedge.

For a comprehensive overview of these constructions, see Geometry Constructions (15 Must Know Types) with Compass and Straightedge.

Copy an Angle​

To copy an angle from one location to another:

  1. Draw a Compass arc (circle) from the vertex of the original angle, intersecting both sides
  2. Draw the same arc (circle) from the new vertex
  3. Use Compass to measure the distance between the two intersection points on the original angle
  4. Transfer this distance to the new arc to locate the second side of the copied angle

Construct a Perpendicular Bisector​

To find all points equidistant from two given points:

  1. Draw a Segment between the two points (A and B)
  2. From point A, draw a Cirlce or a Compass arc (circle) with radius greater than half the segment length
  3. From point B, draw a Compass arc (circle) with the same radius
  4. The arcs intersect at two points
  5. Draw a Line through these two intersection points
  6. This line is the perpendicular bisector — any point on it is equidistant from A and B
tip

In MotionGen, you can use the Perpendicular Bisector tool directly by selecting a segment.

Copy a Segment​

To transfer a length from one segment to another location:

  1. Use Compass with the original segment's endpoints to set the radius
  2. Place the compass center at the new starting point
  3. The circle shows all points at the same distance from the center

Construct a Perpendicular Line Through a Point​

To create a line perpendicular to a given line through a point P:

  1. From point P, draw a Circle or a Compass arc (circle) that intersects the line at two points (A and B)
  2. From A and B, draw two Compass arcs (circle) with equal radius that intersect at point C
  3. Draw a Line through P and C
  4. This line is perpendicular to the original line
tip

In MotionGen, you can use the Perpendicular Line tool directly by selecting a segment or line and a point.

Construct a Parallel Line Through a Point​

To create a line parallel to a given line through a point P:

  1. Draw a Line through P that intersects the original line at point A (creating a transversal)
  2. Copy the angle formed at A to point P using the Copy an Angle construction
  3. The new line through P at this angle is parallel to the original line
tip

In MotionGen, you can use the Parallel Line tool directly by selecting a segment or line and a point.

Bisect an Angle​

To divide an angle into two equal parts:

  1. Draw a Circle or a Compass arc (circle) from the vertex, intersecting both sides of the angle
  2. From each intersection point, draw equal arcs that intersect each other
  3. Draw a line from the vertex through the intersection of the arcs
  4. This line is the Angle Bisector
tip

In MotionGen, you can use the Angle Bisector tool directly by selecting three points that define the angle.

Copy a Triangle Using SAS​

To copy a triangle using Side-Angle-Side (SAS):

  1. Copy the first Side using the Copy a Segment construction
  2. Copy the Angle at one endpoint using the Copy an Angle construction
  3. Copy the second Side along the new angle using Compass
  4. Connect the endpoints to complete the triangle

Copy a Triangle Using SSS​

To copy a triangle using Side-Side-Side (SSS):

  1. Copy the first Side (base) using the Copy a Segment construction
  2. From one endpoint, draw a Compass arc (circle) with radius equal to the second side
  3. From the other endpoint, draw a Compass arc (circle) with radius equal to the third side
  4. The intersection of the two arcs is the third vertex of the triangle
  5. Connect the vertices to complete the triangle

Copy a Triangle Using ASA​

To copy a triangle using Angle-Side-Angle (ASA):

  1. Copy the Side (base) using the Copy a Segment construction
  2. Copy the first Angle at one endpoint using the Copy an Angle construction
  3. Copy the second Angle at the other endpoint using the Copy an Angle construction
  4. The intersection of the two angle lines is the third vertex of the triangle