Accurate upper bound for bezier arc approximation error

A system for rendering text glyphs using circular arcs, the system comprising:

• at least one processor;
• and
• a memory device accessible to the processor, the memory device including processor-readable instructions, which when executed by the processor, configure the processor to perform functions to:
• approximate a first Bezier curve defining a portion of a glyph outline to be rendered on a display of the computing device, based on a second Bezier curve representing a Bezier approximation of a circular arc, the second Bezier curve having the same endpoints as the first Bezier curve;
• estimate an upper bound for an approximation error based on a maximum distance between the first Bezier curve and the second Bezier curve within a two-dimensional coordinate space;
• convert the first Bezier curve to the circular arc approximated using the second Bezier curve for the portion of the glyph outline within the two-dimensional coordinate space, based on the estimated upper bound of the approximation error;
• and
• display the portion of the glyph outline within an area of the display based on the conversion.

A non-transitory machine-readable medium comprising instructions stored therein, which when executed by a system, cause the system to perform operations comprising:

• approximating, by a computing device, a first Bezier curve using a second Bezier curve representing a Bezier approximation of a circular arc, the first Bezier curve defining a portion of a glyph outline to be rendered on a display of the computing device, the second Bezier curve having the same endpoints as the first Bezier curve, and each of the first and second Bezier curves being defined by an error function including one or more parameters related to a shape of the respective first and second Bezier curves;
• transforming each of the first and second Bezier curves so as to be invariant to changes in rotation with respect to a two-dimensional coordinate space corresponding to an area of a display of the computing device to which the glyph outline is to be rendered, the transformed second Bezier curve having reflectional symmetry within the two-dimensional coordinate space;
• determining a maximum distance between the first Bezier curve and the circular arc being approximated by the second Bezier curve within the two-dimensional coordinate space based on a length of a difference vector from each point on the first Bezier curve to a corresponding point on the second Bezier curve, the points on the first and second Bezier curves corresponding to matching parameter values of the respective error functions defining each of the first and second Bezier curves;
• estimating an upper bound for an approximation error between the first Bezier curve and the circular arc based on the determined maximum distance between the first and second Bezier curves in the two-dimensional coordinate space;
• converting the first Bezier curve to the circular arc approximated with the second Bezier curve for the portion of the glyph outline within the two-dimensional coordinate space, based on the estimated upper bound of the approximation error;
• and
• causing the portion of the glyph outline to be rendered to the area of the display represented by the two-dimensional coordinate space, based on the converted first Bezier curve.