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Full Discussion: Returns points of an ellipse
Top Forums Shell Programming and Scripting Returns points of an ellipse Post 302786381 by DGPickett on Wednesday 27th of March 2013 10:49:28 AM
Old 03-27-2013
Not an ellipsoidal solid (an ellipse spun on the line through its axes), then, just an ellipse in a plane.

I guess an ellipse reduced to a unit ellipse is a circle, so not so much scaling as rotation and translation. Do you start with just a few points? Takes three to set the plane, which sets the rotation need, and then I expect a couple more to fix the ellipse in position, size and orientation. Having the axes is so much more concise than having random edge points. Obviously, some points might be redundant in fixing the ellipse, such as 4 points of a rectangle, which the solution must be robust in detecting.

The characteristic of an ellipse is that the reflection angles of rays from one axis hit the other, but trig is processing expensive and not friendly for gauss-jordan reduction. More friendly is that all points have the same sum of distances from the two axes, and are in the surface of a cone. Three-D distance is not friendly, but after rotation it is 2d. The 2d distance calc is polynomial, but with the right scaling, separately for x and y, and translation, the first order terms might be zeroed, so the squares become substitute terms in the GJ. It is reminiscent of deriving the binomial theorem, if you recall back then!

There are colections of solutions for most geometric problems, but doing it yourself is so much nicer. I recall regenerating the binomial theorem while taking a math SAT (got 800), but since then for some reason it has stuck! I never liked the trig in calculus, as there were so many identities to memorize, but most Integral Calculus tests were open book.

Last edited by DGPickett; 03-27-2013 at 11:55 AM..
 

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XDrawArc(3)							  XLIB FUNCTIONS						       XDrawArc(3)

NAME
XDrawArc, XDrawArcs, XArc - draw arcs and arc structure SYNTAX
int XDrawArc(Display *display, Drawable d, GC gc, int x, int y, unsigned int width, unsigned int height, int angle1, int angle2); int XDrawArcs(Display *display, Drawable d, GC gc, XArc *arcs, int narcs); ARGUMENTS
angle1 Specifies the start of the arc relative to the three-o'clock position from the center, in units of degrees * 64. angle2 Specifies the path and extent of the arc relative to the start of the arc, in units of degrees * 64. arcs Specifies an array of arcs. d Specifies the drawable. display Specifies the connection to the X server. gc Specifies the GC. narcs Specifies the number of arcs in the array. width height Specify the width and height, which are the major and minor axes of the arc. x y Specify the x and y coordinates, which are relative to the origin of the drawable and specify the upper-left corner of the bound- ing rectangle. DESCRIPTION
XDrawArc draws a single circular or elliptical arc, and XDrawArcs draws multiple circular or elliptical arcs. Each arc is specified by a rectangle and two angles. The center of the circle or ellipse is the center of the rectangle, and the major and minor axes are specified by the width and height. Positive angles indicate counterclockwise motion, and negative angles indicate clockwise motion. If the magni- tude of angle2 is greater than 360 degrees, XDrawArc or XDrawArcs truncates it to 360 degrees. For an arc specified as [x,y,width,height,angle1,angle2], the origin of the major and minor axes is at [x+_____,y+______], and the infin- itely thin path describing the entire circle or ellipse intersects the horizontal axis at [x,y+______] and [x+width,y+______] and inter- sects the vertical axis at [x+_____,y] and [x+_____,y+height]. These coordinates can be fractional and so are not truncated to discrete coordinates. The path should be defined by the ideal mathematical path. For a wide line with line-width lw, the bounding outlines for filling are given by the two infinitely thin paths consisting of all points whose perpendicular distance from the path of the cir- cle/ellipse is equal to lw/2 (which may be a fractional value). The cap-style and join-style are applied the same as for a line corre- sponding to the tangent of the circle/ellipse at the endpoint. For an arc specified as [x,y,width,height,angle1,angle2], the angles must be specified in the effectively skewed coordinate system of the ellipse (for a circle, the angles and coordinate systems are identical). The relationship between these angles and angles expressed in the normal coordinate system of the screen (as measured with a protractor) is as follows: skewed-angle=atan(tan(normal-angle)*______)+adjust The skewed-angle and normal-angle are expressed in radians (rather than in degrees scaled by 64) in the range [0,2n] and where atan returns a value in the range [-_,_] and adjust is: l l. 0 for normal-angle in the range [0,_] n for normal-angle in the range [_,__] 2n for normal-angle in the range [__,2n] For any given arc, XDrawArc and XDrawArcs do not draw a pixel more than once. If two arcs join correctly and if the line-width is greater than zero and the arcs intersect, XDrawArc and XDrawArcs do not draw a pixel more than once. Otherwise, the intersecting pixels of inter- secting arcs are drawn multiple times. Specifying an arc with one endpoint and a clockwise extent draws the same pixels as specifying the other endpoint and an equivalent counterclockwise extent, except as it affects joins. If the last point in one arc coincides with the first point in the following arc, the two arcs will join correctly. If the first point in the first arc coincides with the last point in the last arc, the two arcs will join correctly. By specifying one axis to be zero, a hori- zontal or vertical line can be drawn. Angles are computed based solely on the coordinate system and ignore the aspect ratio. Both functions use these GC components: function, plane-mask, line-width, line-style, cap-style, join-style, fill-style, subwindow-mode, clip-x-origin, clip-y-origin, and clip-mask. They also use these GC mode-dependent components: foreground, background, tile, stipple, tile-stipple-x-origin, tile-stipple-y-origin, dash-offset, and dash-list. XDrawArc and XDrawArcs can generate BadDrawable, BadGC, and BadMatch errors. STRUCTURES
The XArc structure contains: typedef struct { short x, y; unsigned short width, height; short angle1, angle2; /* Degrees * 64 */ } XArc; All x and y members are signed integers. The width and height members are 16-bit unsigned integers. You should be careful not to generate coordinates and sizes out of the 16-bit ranges, because the protocol only has 16-bit fields for these values. DIAGNOSTICS
BadDrawable A value for a Drawable argument does not name a defined Window or Pixmap. BadGC A value for a GContext argument does not name a defined GContext. BadMatch An InputOnly window is used as a Drawable. BadMatch Some argument or pair of arguments has the correct type and range but fails to match in some other way required by the request. SEE ALSO
XDrawLine(3), XDrawPoint(3), XDrawRectangle(3) Xlib - C Language X Interface X Version 11 libX11 1.5.0 XDrawArc(3)
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