Friday, April 24, 2015

Unizor - Geometry3D - Elements - Angles and Planes

The concept of an angle in general is related to a process of rotation. Thus, if two rays (half-lines) on a plane have a common beginning points, the angle between them can be viewed as an angle needed to rotate one ray around the common point (strictly speaking, counterclockwise is a positive direction and clockwise is negative) to coincide with another.

Provided a unit of measurement of such a rotation (for instance, rotation around a full circle can be set as equal to 360 degree or 2π radian), any angle can be measured in such units.

Talking about an angle between two intersecting lines on a plane, we usually consider the positive smallest angle between half-lines and call it an angle between given lines.

With angles between intersecting planes we will deal analogously. Consider we have two planes α and β that intersect along a line d, which, according to axiom A2 above, is a straight line. We can rotate plane α around line of intersection d until it coincides with plane β. Similarly to the measurement of angles between the lines, we can set the full circle of rotation to 360 degree or 2π radian, thus establishing a measuring unit for angles.

After we introduce other concepts in the corresponding lectures, we will see how measurement of angles between certain lines on the planes corresponds to angles between these planes. Jumping forward, let's just mention without rigorous explanation that, if we construct the third plane γ perpendicular (not yet properly defined) to a line d of intersection of two given planes α and β, the angle between α and β is equal in measurement (degrees or radians) to an angle between two lines lying in the plane γ - a line of intersection of γ with α and a line of intersection of γ with β.

Let's introduce a concept of an angle between a line and a plane. Assume that line d intersects plane α at one point X. Let's draw a line x within plane α through this point of intersection. Depending on the position of line x, the angle between lines d and x will be different. Let's choose the position of line x that gives the smallest value of this angle. This, by definition, is the angle between line d and plane α.
Again, without rigorous explanation that we defer to a future lecture, the line x that produces a smallest angle between line d and plane α is a projection of line d onto plane α that can be constructed by dropping a perpendicular (admittedly, not yet properly defined) from any point on line d outside a plane onto plane α.

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