Examples of Perspective Fields
The hidden structure in Triangle Geometry
Some key points on Perspective Fields
- There exists a general linear relationship between triangle points, referred to here as a Perspective Field.
- A perspective field can be defined by four points, no three of which lie on a straight line.
- Any point within such a field can be specified using perspective coordinates, consisting of three numerical values.
- The perspective coordinates of a point X are the weights associated with the first three defining points of the field.
- The general coordinates of point X are then given by a weighted average of the general coordinates of the first three defining points.
Here are some Perspective Fields (with ETC-points from the range X(1)-X(3500)):
| PF-nr | PF-definition | # pts | List of points |
|---|---|---|---|
| PF1 | PF[X6,X15,X2;X4] | 101 | X3, X4, X5, X6, X13, X14, X15, X16, X17, X18, X20, X30, X61, X62, X140, X371, X372, X376, X381, X382, X395, X396, X397, X398, X485, X486, X546, X547, X548, X549, X550, X590, X615, X631, X632, X1131, X1132, X1139, X1140, X1151, X1152, X1327, X1328, X1587, X1588, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2671, X2672, X2673, X2674, X2675, X2676, X3068, X3069, X3070, X3071, X3090, X3091, X3146, X3311, X3312, X3316, X3317, X3364, X3365, X3366, X3367, X3368, X3369, X3370, X3371, X3372, X3373, X3374, X3379, X3380, X3381, X3382, X3385, X3386, X3387 |
| PF2 | PF[X3,X4,X1;X498] | 53 | X1, X2, X3, X4, X5, X11, X12, X20, X30, X35, X36, X55, X56, X140, X376, X381, X382, X388, X390, X495, X496, X497, X498, X499, X546, X547, X548, X549, X550, X631, X632, X999, X1056, X1058, X1478, X1479, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2675, X2676, X3058, X3085, X3086, X3090, X3091, X3146, X3295, X3303, X3304 |
| PF3 | PF[X2,X4,X1;X40] | 52 | X1, X2, X3, X4, X5, X8, X10, X20, X30, X40, X140, X145, X165, X355, X376, X381, X382, X515, X516, X517, X519, X546, X547, X548, X549, X550, X551, X631, X632, X944, X946, X952, X962, X1125, X1385, X1482, X1483, X1656, X1657, X1698, X1699, X2041, X2042, X2043, X2044, X2045, X2046, X2675, X2676, X3090, X3091, X3146, X3241, X3244 |
| PF4 | PF[X2,X4,X298;X617] | 50 | X2, X3, X4, X5, X20, X30, X69, X140, X298, X299, X302, X303, X376, X381, X382, X487, X488, X489, X490, X491, X492, X546, X547, X548, X549, X550, X616, X617, X621, X622, X627, X628, X631, X632, X633, X634, X637, X638, X1270, X1271, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2675, X2676, X3090, X3091, X3146 |
| PF5 | PF[X6,X15,X1;X35] | 49 | X1, X3, X6, X15, X16, X35, X36, X55, X56, X61, X62, X202, X203, X371, X372, X999, X1124, X1151, X1152, X1250, X1335, X2066, X2067, X2307, X2673, X2674, X3295, X3297, X3298, X3299, X3301, X3303, X3304, X3311, X3312, X3364, X3365, X3368, X3369, X3371, X3372, X3379, X3380, X3385, X3386, X3389, X3390, X3393, X3394, X3395, X3396 |
| PF6 | PF[X2,X4,X6;X182] | 46 | X2, X3, X4, X5, X6, X20, X30, X69, X140, X141, X182, X193, X376, X381, X382, X511, X524, X542, X546, X547, X548, X549, X550, X575, X576, X597, X599, X631, X632, X1350, X1351, X1352, X1353, X1503, X1656, X1657, X1992, X2041, X2042, X2043, X2044, X2045, X2046, X2675, X2676, X3090, X3091, X3098, X3146 |
| PF7 | PF[X2,X4,X6;X32] | 45 | X2, X3, X4, X5, X6, X20, X30, X32, X39, X115, X140, X187, X230, X376, X381, X382, X546, X547, X548, X549, X550, X574, X631, X632, X1285, X1384, X1506, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2548, X2549, X2675, X2676, X3053, X3054, X3055, X3090, X3091, X3146 |
| PF8 | PF[X1,X3,X4;X1737] | 44 | X1, X2, X3, X4, X5, X20, X30, X46, X65, X140, X376, X381, X382, X546, X547, X548, X549, X550, X631, X632, X1155, X1159, X1656, X1657, X1737, X1770, X1788, X1836, X1837, X2041, X2042, X2043, X2044, X2045, X2046, X2646, X2675, X2676, X3090, X3091, X3146, X3474, X3485, X3486 |
| PF9 | PF[X3,X4,X74,X113] | 43 | X2, X3, X4, X5, X20, X30, X74, X110, X113, X125, X140, X146, X265, X376, X381, X382, X399, X541, X542, X546, X547, X548, X549, X550, X631, X632, X1511, X1539, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2675, X2676, X2777, X3090, X3091, X3146, X3448 |
| PF10 | PF[X3,X4,X99,X115] | 43 | X2, X3, X4, X5, X20, X30, X98, X99, X114, X115, X140, X147, X148, X376, X381, X382, X542, X543, X546, X547, X548, X549, X550, X620, X631, X632, X671, X1656, X1657, X2041, X2042, X2043, X2044, X2045, X2046, X2482, X2675, X2676, X2782, X2794, X3090, X3091, X3146 |
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