diff --git a/lib/ephem/computation/chebyshev_polynomial.rb b/lib/ephem/computation/chebyshev_polynomial.rb
index 3ad3101..7fdea25 100644
--- a/lib/ephem/computation/chebyshev_polynomial.rb
+++ b/lib/ephem/computation/chebyshev_polynomial.rb
@@ -52,9 +52,9 @@ def self.evaluate(coeffs, t)
       # @param coeffs [Array<Array<Float>>] Array of coefficients; shape is
       #   [n_terms][3].
       # @param t [Float] The normalized independent variable (in [-1, 1]).
-      # @param radius [Float] The half-length of the time interval (days).
+      # @param radius [Float] The half-length of the time interval (seconds).
       # @return [Array<Float>] The 3-vector derivative (velocity), in units per
-      #   second.
+      #   day.
       def self.evaluate_derivative(coeffs, t, radius)
         n = coeffs.size
         return [0.0, 0.0, 0.0] if n < 2
diff --git a/spec/ephem/segments/chebyshev_type2_spec.rb b/spec/ephem/segments/chebyshev_type2_spec.rb
new file mode 100644
index 0000000..9242cf2
--- /dev/null
+++ b/spec/ephem/segments/chebyshev_type2_spec.rb
@@ -0,0 +1,218 @@
+# frozen_string_literal: true
+
+# Minimal host that mixes in the module and exposes the state its methods
+# read, so the shared evaluation machinery can be exercised on its own.
+class ChebyshevType2TestHost
+  include Ephem::Segments::ChebyshevType2
+
+  attr_reader :midpoints, :radii, :coefficients
+
+  def initialize(component_count: 3, midpoints: nil, radii: nil,
+    coefficients: nil)
+    @component_count = component_count
+    @midpoints = midpoints
+    @radii = radii
+    @coefficients = coefficients
+    @data_lock = Mutex.new
+    @data_loaded = false
+  end
+
+  private
+
+  attr_reader :component_count
+end
+
+RSpec.describe Ephem::Segments::ChebyshevType2 do
+  describe "#time_to_seconds" do
+    it "returns zero at the J2000 epoch" do
+      host = build_host
+
+      expect(host.send(:time_to_seconds, 2451545.0, 0.0)).to eq(0.0)
+    end
+
+    it "scales whole days to seconds past J2000" do
+      host = build_host
+
+      expect(host.send(:time_to_seconds, 2451546.0, 0.0)).to eq(86400.0)
+    end
+
+    it "folds the fractional offset into the result" do
+      host = build_host
+
+      expect(host.send(:time_to_seconds, 2451545.0, 0.5)).to eq(43200.0)
+    end
+  end
+
+  describe "#convert_to_seconds" do
+    it "converts a single time" do
+      host = build_host
+
+      expect(host.send(:convert_to_seconds, 2451546.0, 0.0)).to eq(86400.0)
+    end
+
+    it "converts each element of an array of times" do
+      host = build_host
+
+      result = host.send(:convert_to_seconds, [2451545.0, 2451546.0], 0.0)
+
+      expect(result).to eq([0.0, 86400.0])
+    end
+  end
+
+  describe "#find_interval" do
+    it "selects the interval whose window covers the time" do
+      host = build_search_host
+
+      expect(host.send(:find_interval, 100.0)).to eq(0)
+      expect(host.send(:find_interval, 300.0)).to eq(1)
+    end
+
+    it "treats both ends of an interval window as covered" do
+      host = build_search_host
+
+      expect(host.send(:find_interval, 90.0)).to eq(0)
+      expect(host.send(:find_interval, 110.0)).to eq(0)
+    end
+
+    it "still returns the right interval after caching a previous one" do
+      host = build_search_host
+
+      host.send(:find_interval, 105.0)
+
+      expect(host.send(:find_interval, 105.0)).to eq(0)
+      expect(host.send(:find_interval, 300.0)).to eq(1)
+    end
+
+    it "raises when the time falls in a gap between intervals" do
+      host = build_search_host
+
+      expect { host.send(:find_interval, 200.0) }
+        .to raise_error(Ephem::OutOfRangeError)
+    end
+
+    it "raises when the time is before the first interval" do
+      host = build_search_host
+
+      expect { host.send(:find_interval, 50.0) }
+        .to raise_error(Ephem::OutOfRangeError)
+    end
+
+    it "raises when the time is after the last interval" do
+      host = build_search_host
+
+      expect { host.send(:find_interval, 350.0) }
+        .to raise_error(Ephem::OutOfRangeError)
+    end
+  end
+
+  describe "#time_in_interval?" do
+    it "is true on the inclusive bounds of the window" do
+      host = build_normalized_host
+
+      expect(host.send(:time_in_interval?, 90.0, 0)).to be true
+      expect(host.send(:time_in_interval?, 110.0, 0)).to be true
+    end
+
+    it "is false just outside the window" do
+      host = build_normalized_host
+
+      expect(host.send(:time_in_interval?, 89.0, 0)).to be false
+      expect(host.send(:time_in_interval?, 111.0, 0)).to be false
+    end
+  end
+
+  describe "#compute_normalized_time" do
+    it "maps the midpoint to zero and the window edges to plus/minus one" do
+      host = build_normalized_host
+
+      expect(host.send(:compute_normalized_time, 100.0, 0)).to eq(0.0)
+      expect(host.send(:compute_normalized_time, 110.0, 0)).to eq(1.0)
+      expect(host.send(:compute_normalized_time, 90.0, 0)).to eq(-1.0)
+      expect(host.send(:compute_normalized_time, 105.0, 0)).to eq(0.5)
+    end
+  end
+
+  describe "#process_coefficient_data" do
+    it "splits each record into a midpoint, a radius, and coefficients" do
+      host = build_host(component_count: 2)
+      raw = [
+        100.0, 10.0, 1.0, 2.0, 3.0, 4.0,
+        200.0, 20.0, 5.0, 6.0, 7.0, 8.0
+      ]
+      record_size = 6
+      segment_count = 2
+      coefficient_count = 2
+
+      host.send(
+        :process_coefficient_data,
+        [raw, record_size, segment_count, coefficient_count]
+      )
+
+      expect(host.midpoints).to eq([100.0, 200.0])
+      expect(host.radii).to eq([10.0, 20.0])
+      expect(host.coefficients).to eq(
+        [
+          [[1.0, 3.0], [2.0, 4.0]],
+          [[5.0, 7.0], [6.0, 8.0]]
+        ]
+      )
+    end
+  end
+
+  describe "#generate_position" do
+    it "evaluates the polynomial at the normalized time" do
+      host = build_evaluation_host
+
+      result = host.send(:generate_position, 0.5)
+
+      expect(result).to eq([6.0, 12.0, 18.0])
+    end
+  end
+
+  describe "#generate_single" do
+    it "returns a position identical to the standalone evaluation" do
+      host = build_evaluation_host
+
+      position, = host.send(:generate_single, 0.5)
+
+      expect(position).to eq(host.send(:generate_position, 0.5))
+    end
+
+    it "returns a velocity identical to the standalone derivative" do
+      host = build_evaluation_host
+      expected_velocity = Ephem::Computation::ChebyshevPolynomial
+        .evaluate_derivative(evaluation_coefficients, 0.5, 1.0)
+
+      _position, velocity = host.send(:generate_single, 0.5)
+
+      expect(velocity).to eq(expected_velocity)
+    end
+  end
+
+  def build_host(component_count: 3)
+    ChebyshevType2TestHost.new(component_count: component_count)
+  end
+
+  def build_search_host
+    ChebyshevType2TestHost.new(
+      midpoints: [100.0, 300.0],
+      radii: [10.0, 10.0]
+    )
+  end
+
+  def build_normalized_host
+    ChebyshevType2TestHost.new(midpoints: [100.0], radii: [10.0])
+  end
+
+  def build_evaluation_host
+    ChebyshevType2TestHost.new(
+      midpoints: [0.0],
+      radii: [1.0],
+      coefficients: [evaluation_coefficients]
+    )
+  end
+
+  def evaluation_coefficients
+    [[1.0, 2.0, 3.0], [10.0, 20.0, 30.0]]
+  end
+end