diff --git a/lib/build-trajectory.js b/lib/build-trajectory.js
index e6fbd6c..226d8c4 100644
--- a/lib/build-trajectory.js
+++ b/lib/build-trajectory.js
@@ -11,7 +11,9 @@ const {default: linesIntersection} = require('@turf/line-intersect')
 // returns [distToSegment, nearestPoint]
 const nearestOnSegment = (pt, segPtA, segPtB) => {
 	const distToA = distance({type: 'Point', coordinates: segPtA}, pt)
+	if (distToA === 0) [segPtA, distToA]
 	const distToB = distance({type: 'Point', coordinates: segPtB}, pt)
+	if (distToB === 0) [segPtB, distToB]
 
 	const perpLength = Math.max(distToA, distToB)
 	const dir = bearing(segPtA, segPtB)
@@ -60,6 +62,7 @@ const addInterpolatedTimes = (shapePt, fromI, toI) => {
 	}
 }
 
+// todo: use shape_dist_traveled if available
 const buildTrajectory = async (shapeId, shapePoints, schedule, allStopLocs) => {
 	// We violate the GeoJSON spec by a certain extent here:
 	// > A position is an array of numbers. There MUST be two or more
@@ -92,6 +95,51 @@ const buildTrajectory = async (shapeId, shapePoints, schedule, allStopLocs) => {
 	const arrivals = Array.from(schedule.arrivals)
 	const departures = Array.from(schedule.departures)
 	const stopLocs = await Promise.all(schedule.stops.map(id => allStopLocs.get(id)))
+	const insertedShapePts = new Set()
+
+	// handle 1st stop if before 1st segment
+	const [firstNearest, firstDistance] = nearestOnSegment(stopLocs[0], shapePts[0], shapePts[1])
+	if (firstNearest === shapePts[0] && firstDistance > 0) {
+		// 1st stop is neither closer to 2nd shape pt nor to perpendicular into 1st segment
+		stopIds.shift()
+		const arrival = arrivals.shift()
+		const departure = departures.shift()
+		const stopLoc = stopLocs.shift()
+		const newShapePt = [
+			stopLoc[0], // longitude
+			stopLoc[1], // latitude
+			null, // altitude
+			arrival,
+			departure,
+		]
+		debug('adding helper shape segment', newShapePt, 'at 0')
+		shapePts.unshift(newShapePt)
+		insertedShapePts.add(newShapePt)
+	}
+
+	// handle last stop if after last segment
+	const [lastNearest, lastDistance] = nearestOnSegment(
+		stopLocs[stopLocs.length - 1],
+		shapePts[shapePts.length - 2],
+		shapePts[shapePts.length - 1],
+	)
+	if (lastNearest === shapePts[shapePts.length - 1] && lastDistance > 0) {
+		// last stop is neither closer to 2nd-last shape pt nor to perpendicular into last segment
+		stopIds.pop()
+		const arrival = arrivals.pop()
+		const departure = departures.pop()
+		const stopLoc = stopLocs.pop()
+		const newShapePt = [
+			stopLoc[0], // longitude
+			stopLoc[1], // latitude
+			null, // altitude
+			arrival,
+			departure,
+		]
+		debug('adding helper shape segment', newShapePt, 'at', shapePts.length)
+		shapePts.push(newShapePt)
+		insertedShapePts.add(newShapePt)
+	}
 
 	// iterate over shape segments, identify nearby stops, add arrival/departure times
 	let stopsI = 0
@@ -104,16 +152,32 @@ const buildTrajectory = async (shapeId, shapePoints, schedule, allStopLocs) => {
 		const [
 			nearest, distance,
 		] = nearestOnSegment(stopLoc, fromShapePt, toShapePt)
+		const iNearest = nearest === fromShapePt
+			? i - 1
+			: (nearest === toShapePt ? i : NaN)
 		debug({
-			stopsI, stopId: stopIds[stopsI], stopLoc,
-			i, fromShapePt: shapePts[i - 1], toShapePt: shapePts[i],
-			nearest: nearest === fromShapePt ? 'fromShapePt' : (nearest === toShapePt ? 'toShapePt' : 'intersection'),
+			stopsI, maxStopsI: stopIds.length - 1,
+			stopId: stopIds[stopsI], stopLoc,
+			i, maxI: shapePts.length,
+			fromShapePt: shapePts[i - 1], toShapePt: shapePts[i],
+			iNearest, nearest,
 			distance, prevDistance, prevPrevDistance,
 			iPrevNearest, prevNearest,
 		})
 
-		// Has the shape just approached the stop?
-		if (
+		if (distance === 0) {
+			const shapePt = shapePts[iNearest]
+			debug('adding times to shape pt', shapePt, 'at', iNearest)
+			shapePt[3] = arrivals[stopsI]
+			shapePt[4] = departures[stopsI]
+
+			stopsI++
+			i-- // nearest was 1 iter. ago, repeat loop with *current* segment
+			prevPrevDistance = prevDistance = Infinity
+			prevNearest = null
+			iPrevNearest = NaN
+		} else if (
+			// Has the shape come closest to the stop during the last segment?
 			prevDistance < prevPrevDistance
 			&& distance >= prevDistance
 			&& prevDistance < .3 // 300m
@@ -132,32 +196,26 @@ const buildTrajectory = async (shapeId, shapePoints, schedule, allStopLocs) => {
 				]
 				debug('adding perpendicular as shape pt', newShapePt, 'at', iPrevNearest)
 				shapePts.splice(iPrevNearest, 0, newShapePt)
+				insertedShapePts.add(newShapePt)
 				i++
 			} else {
 				// previous nearest was a shape point
-				debug('adding times to shape pt', shapePts[iPrevNearest], 'at', iPrevNearest)
-				shapePts[iPrevNearest][3] = arrivals[stopsI]
-				shapePts[iPrevNearest][4] = departures[stopsI]
+				const shapePt = shapePts[iPrevNearest]
+				debug('adding times to shape pt', shapePt, 'at', iPrevNearest)
+				shapePt[3] = arrivals[stopsI]
+				shapePt[4] = departures[stopsI]
 			}
 
+			stopsI++
 			i-- // nearest was 1 iter. ago, repeat loop with *current* segment
 			prevPrevDistance = prevDistance = Infinity
 			prevNearest = null
 			iPrevNearest = NaN
-			stopsI++
 		} else {
 			prevPrevDistance = prevDistance
 			prevDistance = distance
-			if (nearest === fromShapePt) {
-				prevNearest = fromShapePt
-				iPrevNearest = i - 1
-			} else if (nearest === toShapePt) {
-				prevNearest = toShapePt
-				iPrevNearest = i
-			} else { // nearest is perpendicular into segment
-				prevNearest = nearest
-				iPrevNearest = NaN
-			}
+			iPrevNearest = iNearest
+			prevNearest = nearest
 		}
 	}
 
@@ -178,6 +236,13 @@ const buildTrajectory = async (shapeId, shapePoints, schedule, allStopLocs) => {
 		}
 	}
 
+	for (let i = 0, l = shapePts.length; i < l; i++) {
+		if (insertedShapePts.has(shapePts[i])) {
+			shapePts.splice(i, 1) // remove it
+			i--
+		}
+	}
+
 	return {
 		type: 'Feature',
 		properties: {