13.2 Interpolating Elevations and Plotting Features

13.2 Interpolating Elevations and Plotting Features

Quick Answer: Interpolate with a horizontal distance ratio between two known elevations. Plot control and breaklines (curb, bank, building) before random shots. A profile is station versus elevation along the alignment; a cross-section is offset versus elevation across it. Worked: 15 ft of a 40-ft run from 152.20 ft toward 153.80 ft is 152.80 ft. On a profile, 20+00 = 410.00 ft and 20+50 = 408.50 ft give 409.40 ft at 20+20.

The 2022 CES Mapping domain asks you to interpolate elevations from topographic data, plot topographical features from field information, and plot profiles and cross-sections. Chapter 12 used contours to build a section for earthwork and interference. This section is the mapping counterpart: take field notes and a contour map, put elevations and features on paper (or in the surface model), and describe the mini-plot you would draw.

Uniform-slope interpolation

Unless the item gives a breakline or a different ground model, assume a uniform slope between adjacent known elevations. For point P between A and B:

h_P = h_A + (d_AP / d_AB) × (h_B − h_A)

Distances d_AP and d_AB are horizontal. Convert map millimeters through the stated scale to ground feet before forming the ratio. The ratio is not a fraction of the printed contour label.

Worked interpolation between spot shots

Two ground shots are 40 ft apart. Elevation at A is 152.20 ft. Elevation at B is 153.80 ft. Point P lies 15 ft from A toward B, with no breakline between them.

d_AP / d_AB = 15 / 40 = 0.375
h_P = 152.20 + 0.375 × (153.80 − 152.20) = 152.20 + 0.375 × 1.60 = 152.20 + 0.60 = 152.80 ft

Traps on this setup:

• Averaging 152.20 and 153.80 to 153.00 ft as if P were halfway. Halfway would be 20 ft, not 15 ft.
• Adding 0.375 ft to 152.20 instead of 0.375 × 1.60.
• Measuring 15 ft from B: 153.80 − 0.60 = 153.20 ft, a different point.
• Using slope distance along the ground instead of horizontal map distance.

Worked interpolation between 2-ft contours

A site map uses a 2-ft contour interval. Point Q lies 3/8 of the way from the 88-ft contour to the 90-ft contour.

h_Q = 88 + (3/8) × 2 = 88 + 0.75 = 88.75 ft

Same formula, smaller interval than a 10-ft corridor map. If you counted from the 90-ft contour down, 3/8 of 2 ft is still 0.75 ft, but the elevation would be 90 − 0.75 = 89.25 ft — only if the stem said you were going from 90 toward 88. Direction is part of the given.

A second contour check: a point 40% of the way from 214.0 to 216.0 is 214.0 + 0.40 × 2.0 = 214.8 ft. That is a ground interpolation. A grate spotted 214.8 is not "the 40% point"; it is a feature elevation you plot as a label, not as a contour.

Do not interpolate across a breakline as if the lawn were continuous. From top of curb to flowline the slope is the curb face, not the parkway. From top of bank to toe the slope is the bank. The formula still works along the breakline between two shots on that same feature.

Plotting field points, breaklines, buildings, and curb

Plotting from field information is a sequence, not a scatter of ticks.

1. Control first. Plot recovered hubs, the BM or TBM, and the basis of bearings or project north. Every later shot hangs on that skeleton.
2. Breaklines next. A breakline is a linear feature where slope changes: top of bank, toe of slope, edge of water, crown, flowline, back of curb, building wall. Contours should kink at breaklines. If you triangulate random shots across a ditch, the surface fills the gutter and the map lies.
3. Buildings by corners. Occupy or side-shot each corner; connect in perimeter order. Do not cross a diagonal and call it a wall. Finished-floor spots belong at the door or a stated corner, labeled FF, not as ground contours through the slab.
4. Curb as a three-line object. Back of curb, top of curb, and flowline are separate 3D lines. A 6-inch curb with TC 412.55 ft and FL 412.05 ft at Sta 14+25, 18.0 ft RT, is two elevations at nearly the same plan position. Plot both. Contours meet the face at the TC elevation along the lip; they do not slice the gutter as if it were a uniform 2% lawn.
5. Random topographic shots last. They fill panels between breaklines so interpolation has something to grab.

Station-offset notes are already in alignment language: Sta, LT or RT, elevation, plus a code (GRD, TC, FL, TOB, BLDG). Rectangular notes are northing, easting, elevation, code. Either set plots. Do not mix LT/RT from one day with a swapped face of the instrument the next day without reducing to the same upstation convention.

Plotting profiles from station-offset-elevation notes

A profile is a vertical slice along the alignment. Horizontal axis is station. Vertical axis is elevation. Civil profiles almost always use a larger vertical scale than the plan (vertical exaggeration) so grades are readable. A 1 inch = 40 ft plan with a 1 inch = 4 ft profile makes a −3% grade look steep. Read the labeled elevations and stations, not the apparent paper slope, when you extract a grade.

Worked mini profile

Existing-ground shots on centerline:

Grade 20+00 to 21+00: (407.00 − 410.00) / 100 ft = −0.030 = −3.00%. The ground falls 3.00 ft in 100 ft. From 21+00 to 22+00 the ground is 407.00 at both ends with a sag to 406.50 at 21+50 — a 0.50-ft dip, not a constant grade. Do not force a single slope through a sag.

Interpolate station 20+20 between 20+00 and 20+50. Horizontal run is 50 ft; 20+20 is 20 ft from 20+00, so the ratio is 20/50 = 0.40.

h = 410.00 + 0.40 × (408.50 − 410.00) = 410.00 + 0.40 × (−1.50) = 410.00 − 0.60 = 409.40 ft

Plot five points on station–elevation paper and connect them. The polyline drops steadily through 21+00, bottoms near 21+50, and rises slightly to 22+00. Proposed finished grade from the improvement-plan profile is a separate polyline on the same axes. Cut and fill are the vertical differences between those two lines, station by station — Domain III earthwork uses that plot; Domain IV asks you to draw it from the notes.

Mini cross-section from the same notes

A cross-section is a vertical slice perpendicular to the alignment at a stated station. Horizontal axis is offset (left negative or labeled LT, right labeled RT). Vertical axis is elevation. Left and right are looking in the direction of increasing station. Swap them and the ditch jumps to the other side of the road.

At Sta 21+00, field notes (existing ground unless coded):

Plot those seven points. The ground falls from 30 LT through centerline, continues down to a flowline at 15 RT, then climbs a bank to 40 RT. That right-side dip is a ditch. The top of bank at 25 RT is a breakline; do not draw a straight chord from the flowline to 40 RT that ignores 407.50. Superimpose the proposed typical section on the same axes when the item asks for catch or hinge; the ground polyline itself is the mapping product from the notes.

If the same station has a proposed FG of 407.00 at CL, the existing CL shot and the design grade coincide there. Offsets still differ: existing 12 RT is 406.40 while a 2% crown on a 12-ft lane would be 407.00 − 0.02 × 12 = 406.76 ft. That 0.36-ft difference is why you plot existing and proposed as two polylines, not one.

Exam traps

• Interpolating across a curb face or ditch as if slope were uniform lawn.
• Using map millimeters in the interpolation ratio without converting through scale.
• Averaging two elevations when the point is not halfway.
• Plotting LT/RT from the south end of the job as if stationing decreased.
• Mixing a profile (along) with a cross-section (across).
• Reading slope off an exaggerated profile grid instead of from labeled stations and elevations.
• Connecting building corners out of perimeter order so the footprint crosses itself.

On Prometric CES items the notes or the figure are in the stem. Use those numbers. Bound textbooks in the allowed box will not replace a 15-ft / 40-ft ratio that is already printed on the screen.