29.5 Axes and Quadrants

Axes and Quadrants is the division of the coordinate plane into four distinct regions by the Horizontal Coordinate Axis and the Vertical Coordinate Axis, together with the classification of every point according to the specific pattern of positive and negative values its two coordinates display, or according to its special position directly upon one of the two axes.

First Quadrant Sign Pattern describes the region of the coordinate plane in which every point has a positive value in First Coordinate Position and a positive value in Second Coordinate Position, located where Positive Horizontal Direction and Positive Vertical Direction overlap, conventionally positioned in the upper right region of the plane.

Second Quadrant Sign Pattern describes the region in which every point has a negative value in First Coordinate Position and a positive value in Second Coordinate Position, located where Negative Horizontal Direction and Positive Vertical Direction overlap, conventionally positioned in the upper left region of the plane.

Third Quadrant Sign Pattern describes the region in which every point has a negative value in First Coordinate Position and a negative value in Second Coordinate Position, located where Negative Horizontal Direction and Negative Vertical Direction overlap, conventionally positioned in the lower left region of the plane.

Fourth Quadrant Sign Pattern describes the region in which every point has a positive value in First Coordinate Position and a negative value in Second Coordinate Position, located where Positive Horizontal Direction and Negative Vertical Direction overlap, conventionally positioned in the lower right region of the plane.

The diagram below labels each of the four quadrants according to its characteristic sign pattern.

Point on the Horizontal Axis describes a point whose value in Second Coordinate Position is exactly zero, placing it directly upon the Horizontal Coordinate Axis itself, at a position determined entirely by its First Coordinate Position value, corresponding to a case of Zero Coordinate Movement along the vertical direction during plotting.

Point on the Vertical Axis describes a point whose value in First Coordinate Position is exactly zero, placing it directly upon the Vertical Coordinate Axis itself, at a position determined entirely by its Second Coordinate Position value, corresponding to a case of Zero Coordinate Movement along the horizontal direction during plotting.

Point at the Coordinate Origin describes the unique point whose value in both First Coordinate Position and Second Coordinate Position is exactly zero, coinciding precisely with the Coordinate Origin Location itself, the single shared reference point from which every other position on the plane is measured.

Axis Points outside Quadrants is the clarifying principle that any point satisfying Point on the Horizontal Axis or Point on the Vertical Axis, including the special case of Point at the Coordinate Origin, does not belong to any of the four quadrants described by First Quadrant Sign Pattern, Second Quadrant Sign Pattern, Third Quadrant Sign Pattern, or Fourth Quadrant Sign Pattern, since a quadrant is defined strictly by both coordinates being genuinely nonzero and sharing a specific pair of signs, a condition that a point lying directly on either axis does not satisfy.