Surface types in point interpolations
In the Trend surface and Moving surface point interpolations, you have to specify a function which will be used to fit a surface through your input points. In the Trend surface operation, one surface will be fitted through all point values in your point map; in the Moving surface operation, a surface will be fitted for each output pixel through weighted point values which are within the limiting distance to this output pixel.
Below the functions and surface types are listed, as well as the absolute minimum number of points that are mathematically required to fit such a surface. You will always need more points than this absolute mathematical minimum to obtain good results.
In general, the use of simple surfaces is preferred, as these will produce the least articial extreme values.
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Plane: |
the surface is a plane; formula: z = a + bx + cy Minimum number of points required: 3 |
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2nd degree Linear: |
the surface is planar but tilted, i.e. first order plane; formula: z = a + bx + cy + dxy Minimum number of points required: 4 |
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2nd degree Parabolic: |
the surface is a second order polynomial surface; formula: z = a + bx + cy + ex2 + fy2 Minimum number of points required: 5 |
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2nd degree: |
the surface is a full second order polynomial surface; formula: z = a + bx + cy + dxy + ex2 + fy2 Minimum number of points required: 6 |
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3rd degree: |
the surface is a third order polynomial surface; formula: z = a + ... + gx3 + hx2y + ixy2 + jy3 Minimum number of points required: 10 |
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4th degree: |
the surface is a fourth order polynomial surface; formula: z = a + ... + kx4 + lx3y + mx2y2 + nxy3 + oy4 Minimum number of points required: 15 |
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5th degree: |
the surface is a fifth order polynomial surface; formula: z = a + ... +px5 + qx4y +rx3y2 + ... + uy5 Minimum number of points required: 21 |
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6th degree: |
the surface is a sixth order polynomial surface z = a + ... + vx6+... Minimum number of points required: 28 |
See also: