Bilinear Interpolation

The rectangular question

The calculator answers a two-dimensional between-corners question. Four known values are attached to the corners of an ordered rectangle, and the page estimates the value at a target x-y coordinate inside it. The result is one interpolated value. The rectangle is a geometric input boundary, not a claim that the underlying phenomenon is physically rectangular or that the four values came from a uniform sensor grid.

The word bilinear describes the method's behavior: it is linear in x when y is held fixed and linear in y when x is held fixed, although the combined expression contains a product of the two normalized positions. The calculator does not fit a general polynomial or inspect neighboring cells. It uses only the four visible corner values, keeping the derivation reproducible.

The target is constrained to the closed rectangle. A point on an edge is valid, and a point at a corner returns that corner's value in exact arithmetic. A point outside the rectangle would be extrapolation, which needs a separate model and is intentionally rejected here.

• One target value is estimated from four rectangle corners.
• The method is linear along each coordinate direction.
• The closed rectangle includes edges and corners but excludes extrapolation.

Boundary, target, and corner fields

The horizontal boundaries are x0 and x1, with x0 required to be less than x1. The vertical boundaries are y0 and y1, with y0 required to be less than y1. The target coordinates are x and y. The corner names encode positions: f00 is at (x0, y0), f10 at (x1, y0), f01 at (x0, y1), and f11 at (x1, y1). This naming must remain attached to the coordinates.

All ten numeric inputs are finite and lie from negative 1,000,000 through positive 1,000,000. Decimal boundary, target, and corner values are allowed. The endpoint ordering rules are relational and are checked in addition to the numeric bounds. A boundary equal to its partner creates a zero denominator, so equal x or y boundaries are rejected rather than averaged.

Use one coordinate convention for the rectangle and one compatible value convention for the corners. The page does not sort corner labels, convert units, account for map projections, or detect whether a grid value is a measurement, color channel, temperature, or abstract sample. The interpretation belongs to the surrounding data description.

• Corner suffixes 00, 10, 01, and 11 encode x-y positions.
• Every field uses the same finite signed numeric bound.
• Do not swap a corner value without also changing its coordinate meaning.

Normalized coordinates s and t

The horizontal fraction is s = (x - x0)/(x1 - x0). It measures the target's progress from the left boundary to the right boundary. The vertical fraction is t = (y - y0)/(y1 - y0), measuring progress from the lower boundary to the upper boundary. For an in-range target and ordered boundaries, each fraction lies from zero through one in exact arithmetic.

At s = 0, the target is on the x0 side; at s = 1, it is on x1. At t = 0, it is on y0; at t = 1, it is on y1. The four products in the formula are weights: (1-s)(1-t) for f00, s(1-t) for f10, (1-s)t for f01, and st for f11. These weights are nonnegative inside the rectangle and sum to one.

The engine validates the boundaries before division and checks both fractions for finite results. It does not clamp s or t to disguise a target that failed the rectangle check. The target-domain validation is what ensures the weights represent interpolation rather than extrapolation.

• s measures horizontal progress and t measures vertical progress.
• The four weights sum to one inside the rectangle.
• Fractions are validated after ordered-boundary and target checks.

The bilinear combination

The result is p = (1-s)(1-t)f00 + s(1-t)f10 + (1-s)t f01 + st f11. The lower edge is first thought of as a linear interpolation from f00 to f10, while the upper edge is a linear interpolation from f01 to f11. The final step interpolates between those two edge results using t. Expanding those stages produces the four-term formula returned by the calculator.

For the default rectangle, s = (2.5 - 0)/(10 - 0) = 0.25 and t = (5 - 0)/(20 - 0) = 0.25. The weights are 0.5625, 0.1875, 0.1875, and 0.0625. With corners 0, 10, 20, and 30, the weighted sum is 0 + 1.875 + 3.75 + 1.875 = 7.5.

Because the weights sum to one and are nonnegative for an interior target, the result lies between the smallest and largest corner values in exact arithmetic. This is a useful sanity check, not a guarantee that the four corner samples are an accurate representation of a real surface.

• The formula is a weighted sum of all four corners.
• The default example returns 7.5.
• Interior weights are nonnegative and add to one.

Corners, edges, and interior examples

At (x0, y0), s = 0 and t = 0, so only f00 has weight one. At (x1, y0), s = 1 and t = 0, so the result is f10. At (x0, y1), s = 0 and t = 1, so the result is f01. At (x1, y1), both fractions are one and the result is f11. These four tests verify the corner labels and zero-weight behavior.

On the lower edge y = y0, t is zero and the formula becomes (1-s)f00 + sf10, which is ordinary linear interpolation between the lower corners. On the upper edge y = y1, it becomes (1-s)f01 + sf11. On the left and right edges, it similarly reduces to vertical interpolation. This edge reduction makes bilinear interpolation easy to reason about without memorizing every term.

At the rectangle midpoint, s = t = 0.5 and all four corners have weight 0.25. The result is their arithmetic average. If the corners are 0, 10, 20, and 30, the midpoint is 15. The default target is not the midpoint, so its 7.5 result demonstrates unequal weights.

• Each corner reproduces its matching f value.
• Edges reduce to one-dimensional linear interpolation.
• The midpoint is the average of the four corners.

Invalid rectangles and target limits

Equal x boundaries make x1 - x0 zero and are invalid. Reversed x boundaries are also rejected because x0 names the lower or left boundary and x1 names the higher or right boundary. The same two rules apply to y0 and y1. Reordering requires moving the corresponding coordinate interpretation and corner labels together, not just changing one number.

A target x below x0 or above x1 is outside the horizontal interval. A target y below y0 or above y1 is outside the vertical interval. The engine rejects either condition even if the other coordinate is valid. This ensures that the returned value is an interpolation in a rectangle, not a partly extrapolated estimate.

Corner values may be negative, zero, or positive within their bounds. They do not have to be ordered around the rectangle. A saddle-like pattern is allowed mathematically; the weighted result still follows the four supplied samples. The page does not classify the surface shape or reject a negative corner.

• Both x and y boundary pairs must be strictly ordered.
• Both target coordinates must be inside the closed rectangle.
• Corner values may vary in any signed order within their bounds.

Interpretation and common mistakes

The result is the value implied by a locally bilinear surface through the four corner samples. It is not a physical law, a fitted uncertainty model, or a guarantee about values between sparse measurements. Strong curvature, discontinuities, sharp peaks, and features smaller than the rectangle can make the estimate poor even when the arithmetic is exact.

Common mistakes include swapping f01 and f10, treating the first digit as the y position instead of the x position, using a target fraction measured from the wrong boundary, and omitting one of the four weighted terms. Another mistake is applying a corner average everywhere. The average is correct only at the midpoint; other targets require the unequal weights.

To check the implementation, test all four corners, all four edges, and the midpoint. Also confirm that a surface whose corner values follow p = alpha + beta x + gamma y returns that same plane at interior targets. The page does not run a separate surface fit, but these identities provide strong arithmetic checks.

• Corner naming and weight placement are the main bookkeeping risks.
• A corner average is valid only at the rectangle midpoint.
• Corner, edge, midpoint, and planar-surface checks expose formula errors.

Limits, privacy, and safety

This calculator is a bounded four-corner interpolation kernel. It does not perform bicubic or spline interpolation, select a cell from a large grid, interpolate across irregular coordinates, estimate gradients, or extrapolate beyond the rectangle. It also does not convert coordinate units or identify what the corner values represent. Those additional tasks need explicit data and model contracts.

Grid values may encode private maps, sensor readings, financial surfaces, health measurements, or proprietary images. Enter only the ten values needed for the calculation and do not add names, identifiers, credentials, or raw datasets. Share the result with enough coordinate context to avoid confusion but no more sensitive source detail than necessary.

Do not use a bilinear result alone as a medical threshold, engineering safety value, control input, compliance number, or other high-consequence decision. Validate sampling density, units, uncertainty, discontinuities, and domain rules with appropriate software and review. This page is an educational and transparent arithmetic tool.

• The kernel uses one rectangle and four corners only.
• Protect sensitive grid and surface data.
• High-consequence values require sampling and domain validation.

Frequently asked questions

Why are there four corner values? A target inside a rectangle has both a horizontal and a vertical position. The four weighted terms represent the combinations of lower or upper y and left or right x. Using only two corners would define a line, not a two-dimensional bilinear surface. The four values are the minimum corner contract for this method.

Can the result exceed all four corner values? Not for a valid in-range target with finite ordinary weights, because the weights are nonnegative and sum to one. A result outside the corner range suggests a formula error, an out-of-range target, or a different method such as extrapolation. Floating-point display can create tiny rounding differences at boundaries but not a meaningful overshoot under this contract.

What if a corner is missing? The calculator requires all four f fields and does not invent or skip a value. A missing corner needs a separately defined imputation or data-repair method. Supplying zero as a placeholder changes the surface and should be done only when zero is the actual corner value.

• Four corners are required for the two-dimensional weighting.
• Valid in-range results stay within the corner-value range mathematically.
• Missing data need a separate explicit imputation rule.

Reproducibility, precision, and surface judgment

A bilinear result should be recorded with the rectangle boundaries, target coordinates, corner labels, normalized fractions, weights, and final value. The corner labels matter as much as the numbers because f10 and f01 occupy different positions. A compact final scalar without this map cannot show whether the upper-left and lower-right samples were exchanged. Keeping the intermediate weights visible makes the result auditable and supports an independent recomputation.

The strongest basic checks are geometric. At each of the four corners, the matching corner value must be returned. Along the lower edge, the formula must reduce to interpolation between f00 and f10; along the upper edge, it must reduce to interpolation between f01 and f11. The left and right edges provide the corresponding vertical checks. At the midpoint, all four weights are one quarter and the result is the corner average.

The weights have a useful convex-combination interpretation inside a valid rectangle. Each weight is nonnegative and the four weights sum to one. Consequently, the result lies between the smallest and largest corner values in exact arithmetic. This is a sanity check for sign and placement errors, not proof that the estimated surface is accurate. A source with sharp variation can still make a convex weighted result a poor representation of the interior.

The method is linear in each coordinate when the other coordinate is held fixed. It can reproduce a plane exactly because a plane has the required linear behavior in both directions. It does not reproduce every curved surface exactly, and the mixed x-y term in a true bilinear surface may matter. Do not replace the formula with a simple average or a one-dimensional interpolation unless the target lies on the corresponding edge or the model explicitly calls for that simplification.

Boundary validation happens before normalized division. Equal x boundaries or equal y boundaries would create a zero denominator, while reversed boundaries would make the field names ambiguous. A target outside either interval is rejected rather than clamped. Clamping a target into the rectangle could return a finite value, but it would hide the fact that the caller asked for extrapolation or supplied an invalid coordinate.

Corner values can be negative, zero, or positive and do not need to be ordered. A saddle pattern is valid under the arithmetic, and the weighted output still follows the four samples. What matters is that every value is finite, belongs to the declared coordinate corner, and uses a compatible unit or sample convention. The page does not inspect a larger grid, identify discontinuities, or impute a missing corner.

Precision can be sensitive near a boundary when a target is represented as a decimal that is almost, but not exactly, equal to the boundary. Use the numeric result and a stated tolerance for comparisons rather than string equality. Do not round the normalized fractions before applying the weights if the value will feed another calculation. The renderer may shorten the display, but the engine keeps the finite calculation result available.

For data and graphics work, also document whether the rectangle is a physical cell, a chart region, an image patch, or an abstract interpolation exercise. Coordinates that look rectangular can represent a transformed map or unequal physical scales, in which case the ordinary bilinear model may need adjustment. The calculator intentionally reports the result under its declared Cartesian rectangle, leaving sampling theory, units, uncertainty, and domain review to the surrounding workflow.

A responsible handoff includes the original four samples, the target, s and t, all four weights, the output, and any known reason to prefer bilinear interpolation over a richer method. Avoid placing private map coordinates, sensor identifiers, customer values, or raw datasets in shared notes. The formula is transparent and local, but the source data and the decision based on the estimate may still be sensitive or consequential.

When the four values come from a grid, document the cell-selection rule and whether the axes are evenly spaced. This calculator uses the actual boundary coordinates, so equal spacing is not required, but a caller must still select the correct neighboring cell. A target close to a cell boundary can legitimately have very different weights in adjacent cells when the corner samples change. That is a property of the data, not a reason to average the two cell results without a separate continuity rule.

The two-stage edge interpretation offers a practical debugging method. Interpolate f00 to f10 along the lower edge and f01 to f11 along the upper edge using s. Then interpolate those two edge values using t. Repeat in the other order by interpolating vertically first and then across x. Both paths should agree for the bilinear formula. This check catches swapped corner labels and incorrect use of s or t more effectively than checking only the final interior example.

A bilinear surface may have a nonzero mixed change even when each edge looks simple. If the four corners are arranged so opposite differences disagree, the interior responds to both normalized coordinates. This is why the four weighted terms cannot generally be replaced by separate averages of rows or columns. The edge reductions remain valid, but the interior needs the full two-dimensional combination.

If corner values are measurements with different confidence levels, ordinary bilinear weights do not express those confidence levels. A weighted fit or uncertainty-aware interpolation may be more appropriate, but it would be a different model. The page gives geometric position weights only. Do not describe a large corner weight as proof that the corresponding measurement is more trustworthy; it only says that the target is closer in the normalized rectangle.

For a visual surface, keep the coordinate axes and value scale visible with the estimated point. A color or height display can hide the fact that a target lies near an edge or that all four corner values are close. The numeric weights provide a more direct audit trail. If an estimate drives a threshold, preserve the full-precision value and the threshold tolerance rather than reading a color bucket as an exact number.

The interpolation is continuous across a shared edge when adjacent cells use the same edge values and compatible coordinate boundaries. This property can be checked by evaluating the edge from both cells and comparing the results. It is not automatic for a data set with mismatched samples, a discontinuity, or a cell-selection error. The calculator evaluates one rectangle and leaves multi-cell continuity to the caller.

A target near a corner is dominated by one corner weight, but the other weights still document how the estimate was formed. Do not discard small weights before calculation because they can matter when corner values differ greatly. Likewise, do not round s and t to a coarse grid unless that quantization is part of the declared application model.

If the value surface represents a rate, concentration, temperature, or another measured quantity, confirm that linear mixing is sensible for that quantity. Some variables are naturally transformed before interpolation, and some cannot be averaged directly. The calculator makes no such transformation because its contract is the explicit bilinear weighted sum of the four entered values.

For a release review, test one affine plane, one constant surface, one edge-only target, one corner target, a signed saddle pattern, and invalid boundary and target cases. Combine those numeric checks with the source-data review and the article's model limits. Passing finite arithmetic tests establishes the kernel behavior, not the adequacy of a larger sampling or visualization workflow.

• Store corner labels, fractions, weights, and output together.
• Use corner, edge, midpoint, and planar-surface identities.
• Use the convex range check as a sanity check, not an accuracy guarantee.
• Reject invalid boundaries and out-of-rectangle targets instead of clamping.
• Assess sampling, units, precision, privacy, and model suitability outside the kernel.